Theory and Applications of Graphs Theory and Applications of Graphs

We present an algorithm to test whether a given graphical degree sequence is forcibly biconnected. The worst case time complexity of the algorithm is shown to be exponential but it is still much better than the previous basic algorithm for this problem. We show through experimental evaluations that the algorithm is eﬃcient on average. We also adapt the classic algorithm of Ruskey et al. and that of Barnes and Savage to obtain some enumerative results about forcibly biconnected graphical degree sequences of given length n and forcibly biconnected graphical partitions of given even integer n . Based on these enumerative results we make some conjectures such as: when n is large, (1) the proportion of forcibly biconnected graphical degree sequences of length n among all zero-free graphical degree sequences of length n is asymptotically a constant C (0 < C < 1); (2) the proportion of forcibly biconnected graphical partitions of even n among all forcibly connected graphical partitions of n is asymptotically 0.


Introduction
We consider graphical degree sequences of finite simple graphs (i.e.finite undirected graphs without loops or multiple edges) where the order of the terms in the sequence does not matter.As such, the terms in a graphical degree sequence are often written in non-increasing order for convenience.An arbitrary non-increasing sequence of non-negative integers a 1 ≥ a 2 ≥ • • • ≥ a n can be easily tested whether it is a graphical degree sequence by using the Erdős-Gallai criterion [5] or the Havel-Hakimi algorithm [7,6].Sierksma and Hoogeveen [12] later summarized seven equivalent criteria to characterize graphical degree sequences.A zero-free graphical degree sequence is also called a graphical partition.The former terminology is often used when the length of the sequences under consideration is fixed while the later is often used when the sum of the terms in the partitions under consideration is fixed.
Enumerating all unlabeled graphs having the same vertex degree sequence and exploring their properties have been of interest.To our knowledge, an efficient algorithm of Meringer [9] is available for the case of regular graphs, but no efficient algorithm is known to solve the problem for general graphical degree sequences.When considering all realizations of the same vertex degree sequence, two notions are very useful.Let P be any property of graphs (e.g.biconnected, critical, Hamiltonian, planar, etc).A graphical degree sequence ) is called potentially P-graphic if it has at least one realization having the property P and forcibly P-graphic if all of its realizations have the property P [10].In [14] we have presented an efficient algorithm to test whether a graphical degree sequence d is forcibly connected and also outlined an algorithmic framework to test whether d is forcibly k-connected for every fixed k ≥ 2. In this paper we will present a more efficient algorithm to test forcibly biconnectedness of graphical degree sequences.Recall that Wang and Kleitman [13] have given a simple characterization of potentially k-connected graphical degree sequences of length n, which takes O(n) time given any input d.However, testing forcibly k-connectedness appears to be much harder.Some sufficient conditions are known for testing forcibly connectedness or forcibly k-connectedness [3,2,4], but they have also been shown to be unnecessary conditions.
In the rest of this paper we review the basic algorithm and present the improved algorithm to characterize forcibly biconnected graphical degree sequences and justify its correctness in Section 2. We analyze the complexity of the algorithm in Section 3. In Section 4 we demonstrate the efficiency of the algorithm through some computational experiments and then present some enumerative results regarding forcibly biconnected graphical degree sequences of given length n and forcibly biconnected graphical partitions of given even integer n.Based on these enumerative results we make some conjectures about the relative asymptotic behavior of considered functions and the unimodality of certain associated integer sequences.We also disprove a conjecture posed in [14] in this section.Finally we conclude in Section 5.

The decision algorithm for forcibly biconnectedness
In this section we first review the basic algorithm to test forcibly biconnectedness of graphical degree sequences proposed in [14] and then present an improved and much more efficient version of the algorithm.We will give a detailed proof why it works and also comment on certain implementation issues.

Review of the basic algorithm
The basic algorithm from [14] to test forcibly biconnectedness of graphical degree sequences is reviewed in Algorithm 1.The idea is simple as follows.First we need to make sure that the input graphical degree sequence d is potentially biconnected and forcibly connected before we continue the test of forcibly biconnectedness.This is why we have lines 1 and 2.
After we confirmed this is the case, we can conclude that the input d is forcibly biconnected as long as we become sure that in every realization of d there is no cut vertex based on the definition of biconnected (non-separable) graphs.In order to simplify our discussion, we call the degree of a cut vertex in a connected graph a cut degree.Recall that we have defined in [14] that a generalized Havel-Hakimi (GHH) operation on a graphical degree sequence d means selecting an arbitrary term d i to remove from d and then selecting an arbitrary collection d S of size d i from the remaining sequence d − {d i } and decrementing each of them by 1.The resulting sequence is notationally written as d = GHH(d, d i , d S ).If d is a non forcibly connected graphical degree sequence under some choice of d i and d S , then the d i is a cut degree and d is not forcibly biconnected.This is because if d is non forcibly connected, i.e. it has a disconnected realization, then d has a connected realization with a cut vertex of degree d i , which shows d is not forcibly biconnected.If no such choice of d i and d S exists, then no cut degree exists and d is forcibly biconnected.As indicated in [14] if d obtained on line 4 is not a graphical degree sequence, then the for loop continues to iterate without returning False on line 6.
We remark that although Algorithm 1 works, its performance is probably poor due to the large number of possible choices of d i and d S in the for loop from lines 3 to 6. See Section 3 for more detailed worst case time complexity analysis.Our rudimentary implementation of Algorithm 1 can start to encounter bottlenecks for input sequences d of length 30 to 40.
Algorithm 1: basic version pseudo-code to test whether a graphical degree sequence is forcibly biconnected (See text for the description of GHH operation).From [14, Algorithm 2] Input: A zero-free graphical degree sequence

The improved algorithm and the proof of its correctness
Our strategy to improve Algorithm 1 is to replace the for loop from lines 3 to 6 with more sophisticated methods to find a potential cut degree.The idea is still simple.Assume there is a cut degree in the input sequence d.Then we can construct two graphical degree sequences from d with total length n + 1 of which n − 1 terms come directly from d and the remaining two constitute a partition of the cut degree.We actually first test some necessary conditions for the existence of a cut degree, during which process an auxiliary set S consisting of the orders of potential smaller induced subgraphs (explained below) is constructed should a cut degree exist.The pseudo code of the improved version is shown in Algorithm 2. The conditional test on line 3 works because d 2 + d n < n is a necessary condition for the existence of a cut degree.
Proof: Assume there is a cut vertex v c in a realization G of d.There are two possible situations with respect to the degree deg(v c ) of v c .Here the vertex degrees are all with respect to G.) Note that there are two possibilities regarding which induced subgraph v 2 and v x belong: (1) . It is irrelevant to us here which of the two occurs.Since v 2 and v x share at most one neighbor in Lines 5 to 8 work as follows.If there is a cut vertex v c in a realization G of d whose removal results in two disjoint induced subgraphs G 1 and G 2 , then there are some restrictions on the order of the smaller of these two subgraphs given d.Without loss of generality we assume from now on |V (G 2 )| ≤ |V (G 1 )| so that G 2 is always the smaller of the two disjoint subgraphs.This piece of code is trying to find out the set S of potential values of |V (G 2 )|.Notice that G 2 must contain at least d n vertices because if it contains at most d n − 1 vertices, then each of its vertices will have a degree at most d n −1 in G, which contradicts the fact that d n is the smallest degree.Obviously G 2 cannot contain more than (n − 1)/2 vertices since G 2 is the smaller of the two induced subgraphs separated by v c .This explains the for loop lower bound (d n ) and upper bound ( (n − 1)/2 ) on line 6.For an integer s between d n and (n − 1)/2 to be a potential order of G 2 (i.e.|V (G 2 )| = s) we must have d n−s+1 ≤ s since all the vertices in G 2 have degrees at most s in G, which makes it necessary for d to have at least s terms that are at most s.The s smallest terms of d are Therefore all the vertices in G 1 have degrees at most n − s − 1 in G. Now other than the requirement that d has at least s terms that are ≤ s, the other necessary condition for d to contain a cut degree is the existence of an additional n − s − 1 terms that are at most n − s − 1.Since s ≤ n − s − 1 based on the chosen lower and upper bounds for s, we must have d 2 ≤ n − s − 1 as the other necessary condition.
The functionality of lines 9 to 10 is now clear.If no potential smaller induced subgraph order |V (G 2 )| can be found in the range from d n to (n − 1)/2 , then definitely no cut degree in d can possibly exist and d must be forcibly biconnected.
After testing the necessary conditions the double for loop from lines 11 to 18 tries to find out if a cut degree d in d exists such that the smaller subgraph G 2 after cut has order s from S.
The conditional test on line 13 is to ensure that the largest degree d * in d that is not the cut degree d does not exceed the order n − s − 1 of the larger subgraph G 1 after cut, should a cut degree d exists.This is because any vertex in G 1 can only have the vertices in G 1 or the cut vertex with degree d as its neighbors in G.If the vertex with the degree d * occurs in G 2 , then d * actually should not exceed the order s of G 2 .However, since s ≤ n − s − 1 due to the upper bound (n − 1)/2 of s on line 6, it is automatically true that d * does not exceed n − s − 1 when it does not exceed s.Now each Lines 14 to 16 construct the sub sequence d L of d consisting exactly of those elements at most the order s of the smaller subgraph G 2 .The motivation for this construction is that the s degrees of the vertices of G 2 must all come from d L .Note that if the cut degree d ≤ s, then we need to remove one copy of d from d L since the cut vertex itself is not considered part of G 2 .This explains why we have line 16.
The conditional test on line 17 is to ensure that there are enough degrees ≤ s from d for G 2 .The sequence d L constitutes the pool of degrees we can select for the vertex degrees of G 2 .Since G 2 has order s and each of its vertices has degree ≤ s, we need d L to have at least s elements.
Line 18 performs exhaustive enumerations to find out if there are two graphical degree sequences s H and s L , which are the vertex degree sequences of respectively, should a cut degree d exists such that the smaller subgraph G 2 is of order s after cut.The vertex degree sequence s L of G − V (G 1 ) consists of the s degrees of the vertices of G 2 in G together with the degree d equal to the number of adjacencies of the cut vertex with G 2 .The vertex degree sequence One thing to note about Algorithm 2 is that the sequences mentioned from line 15 to line 18 are all treated as multisets so that the set difference and set union operations therein should be implemented as multiset operations.We also note that Algorithm 2 performs a test of whether any graphical sequence is forcibly connected or not at most once (on line 1), while Algorithm 1 possibly performs such a test many times (on lines 1 and 5).This is a supplementary explanation of why Algorithm 2 performs much better than Algorithm 1 on many inputs.More detailed run time analysis is presented later.

Extensions of the algorithm
In this section we briefly discuss how to extend Algorithm 2 to perform the additional task of listing all possible cuts of a potentially biconnected and forcibly connected but non forcibly biconnected graphical degree sequence.We also show that the idea of Algorithm 2 to find a cut degree in deciding forcibly biconnectedness can be extended to test forcibly triconnectedness of a graphical degree sequence and beyond.

Enumeration of all possible cuttings
It is easy to see that if the input d is potentially biconnected and forcibly connected but not forcibly biconnected, we can enumerate all possible cuttings (the cut degree, the numbers of adjacencies of the cut vertex to G 1 and G 2 and the degrees of the vertices of G 1 and G 2 in G using the notation of Section 2.2).We can simply report such a cutting on line 18 of Algorithm 2 without returning False at the moment.Such an enumerative algorithm to find all possible cuttings of the input d can be useful when we want to explore the possible realizations of d and their properties.

Testing forcibly k-connectedness of d when k ≥ 3
The idea to find a cut degree in Algorithm 2 can be extended to find a pair of cut degrees for the case of k = 3 and a triple of cut degrees for the case of k = 4, and so on.Now we need to consider more complicated situations about whether those cut vertices with potential cut degrees are themselves adjacent or not, besides the numbers of adjacencies of each of these cut vertices to G 1 and G 2 and the potential orders of G 1 and G 2 .However, we believe a careful implementation should still have better performance than the basic version to test forcibly k-connectedness presented in [14, Algorithm 3].

Complexity analysis
In this section we give an approximate and conservative analysis of the worst case run time performance of Algorithms 1 and 2. We also comment on the computational complexity of the decision problem of testing forcibly biconnectedness of graphical degree sequences.
Let us first consider the basic Algorithm 1.The test of potentially biconnectedness on line 1 can be performed in linear time using the Wang and Kleitman characterization [13].The test of forcibly connectedness on line 1 has been analyzed in [14] with worst case run time probably exponential (around poly(n) n n/2 where poly(n) is some polynomial of n).The number of iterations in the for loop from lines 3 to 6 depends on the number of possible choices of d i and d S .The number of choices for Let us now turn to the improved Algorithm 2. The worst case run time of line 1 to test potentially biconnectedness and forcibly connectedness is about poly(n) n n/2 as indicated above.
It is easy to see that lines 3 to 10 take linear time.The body of the double for loops from line 13 to 18 will be iterated at most O(n 2 ) times since we have O(n) distinct d in d and less than n/2 elements in S. Lines 14 to 16 clearly take O(n) time.Thus, the total run time excluding the test of forcibly connectedness on line 1 and the exhaustive enumeration on line 18 is O(n 3 ).Now we consider line 18.The length of the sequence d L could be up to n and the length s of the sequence s L could be up to (n − 1)/2 .Then the maximum number of all possible choices of s L from d L will be up to n (n−1)/2 , which is exponential.The number of choices for d is O(n).Once the choices for s L and d have been made, the two sequences s L and s H can be constructed and tested whether they are graphical in O(n) time.This shows that the worst case total run time of line 18 is about poly(n) n n/2 .
The most time consuming parts of Algorithm 2 are the testing of forcibly connectedness on line 1 and the exhaustive enumeration on line 18.In the worst case both of these parts can take exponential time.The combined worst case total run time of Algorithm 2 is then about poly(n) n n/2 , which can be seen to be better than the worst case run time of Algorithm 1 by a factor of n n/2 .Suitable data structures can be established for those multiset sequences mentioned on line 18 to avoid duplicate exhaustive enumerations.However, a large number of exhaustive enumerations on line 18 can happen before returning for some inputs based on our computational experiences.
As for the computational complexity of the problem of deciding forcibly biconnectedness of arbitrary graphical degree sequences, it is known to be in co-NP as indicated in [14].However we do not know if it is co-NP-hard or if it is inherently harder than the problem of deciding forcibly connectedness of arbitrary graphical degree sequences.In general one can be interested in whether the problem of deciding forcibly k-connectedness is inherently harder than the problem of deciding forcibly (k − 1)-connectedness for any fixed k.

Computational results
In this section we will first present some experimental performance evaluation results of Algorithm 2 on randomly generated graphical degree sequences.We will then provide some enumerative results about forcibly biconnected graphical degree sequences of given length and forcibly biconnected graphical partitions of a given even integer.Based on these enumerative results we will make some conjectures about the relative asymptotic behavior of related functions and the unimodality of certain associated integer sequences.
Before we continue we remark that Algorithm 1 can only satisfactorily process inputs with length below 50 most of the time unless the input is not potentially biconnected or it can be easily determined to be non forcibly connected.In contrast, Algorithm 2 can handle a much larger range of input lengths.

Performance evaluations of Algorithm 2
Previously in [14] we evaluated our algorithm to test forcibly connectedness of graphical degree sequences using randomly generated long inputs with length up to 10000.In anticipation of the greater challenge to test forcibly biconnectedness than forcibly connectedness, we decide to evaluate Algorithm 2 using randomly generated inputs with length up to 1000.
We adopt a similar evaluation methodology as in [14].Choose a constant p l in the range [0.002,0.49]and a constant p h in the range [p l +0.01,0.99]and generate 100 random graphical degree sequences of length n with largest term around p h n and smallest term around p l n.For each chosen length n, the smallest chosen p l is slightly adjusted to make sure that the smallest term in each generated random input graphical degree sequence is at least 2 since any graphical degree sequence with the smallest term 1 is not potentially biconnected so the answer is always False for such inputs.The largest p l is chosen to be 0.49 because any graphical degree sequence of length n with smallest term at least 0.5n is not only forcibly connected (see [14]) but also forcibly biconnected (see line 3 of Algorithm 2).The constant p h is chosen to be less than 1 so that the largest term in any randomly generated input graphical degree sequence is at most n − 1.
We implemented our Algorithm 2 using C++ and compiled it using g++ with optimization level -O3.The experimental evaluations are performed on a common Linux workstation.We run the code on the randomly generated instances and record the average performance and note the proportion of them that are forcibly biconnected.Table 1 lists the tested p l and p h for the input length n = 500.The input lengths that are chosen to be tested are n = 20, 30, • • • , 100, 200, • • • , 1000.The chosen p l and p h for different input length n are all similar to the case for n = 500.0.44,0.48,0.52,0.53,0.54,0.55,0.56,0.6,0.7,0.8,0.9 0.49 0.491,...,0.499,0.5,0.501,0.502,0.504,0.506,0.508,0.51,0.52,...,0.6,0.7,0.8,0.9 We summarize our experimental evaluation results as follows.1.For all input length n below 100, all instances can be decided instantly (run time < 0.1s).
2. Starting from n = 100 up to n = 1000, certain inputs start to cause Algorithm 2 to run slowly (run time from a few seconds to a few hours or time out).The longer the length n, the more time these difficult instances might take and the higher percentage of these difficult instances occupy in all 100 input instances for a given triple of (n, p l , p h ).In detail, difficult instances could occur when (1) p l is very small (say from 0.005 to 0.05) and p h is very large (say from 0.8 to 0.95) or when (2) p l is slightly below 0.5 and p h is slightly above 0.5.We note based on our observations that the former situation (1) often (but not always) corresponds to the cases where forcibly connectedness itself is hard to decide while the latter situation (2) corresponds to the cases where forcibly connectedness itself is not hard to decide but forcibly biconnectedness is after confirmation of forcibly connectedness.
3. For each fixed p l , there is a transition range of p h (shown in bold in Table 1) such that (1) if p h is below this transition range, almost all input instances are non forcibly biconnected; (2) if p h is above this transition range, almost all input instances are forcibly biconnected; (3) when p h increases in the transition range, the proportion of input instances that are forcibly biconnected grows approximately from 0 to 1.For example, for input length n = 500 and p l = 0.1, the transition range of p h is from 0.68 to 0.8.When p h is below 0.68 almost all input instances are non forcibly biconnected.When p h is above 0.8 almost all input instances are forcibly biconnected.As noted in [14], these results indicate relative frequencies, instead of absolute law, of forcibly biconnected graphical degree sequences among the pool of all graphical degree sequences that can be chosen to test.
To sum up, our implementation of Algorithm 2 runs fast on the majority of the tested random inputs with length up to 1000.Considering that all those inputs with smallest term exactly 1 or at least n/2 are excluded from test, which are estimated to occupy at least 35% of all possible instances, we are confident that Algorithm 2 is efficient on average.Certain inputs do cause it to perform poorly.In particular, if it is hard to decide whether a potentially biconnected input is forcibly connected using the algorithm in [14], it is also hard to decide whether it is forcibly biconnected using Algorithm 2 since testing forcibly connectedness is part of testing forcibly biconnectedness.We also remark from our experimental observations that there are difficult input instances whose hardness come mainly from deciding forcibly connectedness.That is, the algorithm may take a long time to decide forcibly connectedness.However, once the algorithm knows the input is potentially biconnected and forcibly connected it can almost instantly finish the testing of forcibly biconnectedness.There are also difficult input instances that are easy to be determined to be potentially biconnected and forcibly connected but hard to be further determined to be forcibly biconnected.Finally, there are some difficult input instances that are hard to be determined to be forcibly connected and, after confirmation of forcibly connectedness, hard to be further determined to be forcibly biconnected.number of zero-free graphical sequences of length n D k c (n) number of potentially k-connected graphical sequences of length n D k f (n) number of forcibly k-connected graphical sequences of length n C k (n, N ) number of potentially k-connected graphical degree sequences of length n with degree sum N (Note: ) number of forcibly k-connected graphical degree sequences of length n with degree sum N (Note: number of graphical partitions of even n g k c (n) number of potentially k-connected graphical partitions of even n g k f (n) number of forcibly k-connected graphical partitions of even n c k (n, j) number of potentially k-connected graphical partitions of even n with j parts (Note: c k (n, j) = C k (j, n)) f k (n, j) number of forcibly k-connected graphical partitions of even n with j parts (Note: f k (n, j) = F k (j, n)) l k (n, j) number of forcibly k-connected graphical partitions of n with largest term j m k (n) minimum largest term of forcibly k-connected graphical partitions of n

Enumerative results
In this section we present some enumerative results.The motivation to obtain these results is to gain insights into the absolute and relative asymptotic orders of the relevant functions.
For the reader's convenience, we list the notations and their meanings used in this section in Table 2.
In a previous manuscript [15] we presented efficient algorithms to compute D(n) and D k c (n) for fixed k and have shown that lim n→∞ Recently in [14] we presented enumerative results about the number D 1 f (n) of forcibly connected graphical degree sequences of length n and conjectured that lim n→∞ which will be refuted later in Section 4.3.Using the sandwich theorem of calculus we can easily see that lim n→∞ We adapted the algorithm of Ruskey et al [11] that enumerates zero-free graphical degree sequences of length n by incorporating the test in Algorithm 2 to enumerate those that are forcibly biconnected.The results together with the proportion of them in all forcibly connected graphical degree sequences of length n and in all zero-free graphical degree sequences of length n are listed in Table 3.It looks likely that D 2 f (n)/D 1 f (n) and D 2 f (n)/D(n) both tend to some constant less than 1.
Table 3: Number of forcibly biconnected graphical sequences of length n and their proportions in forcibly connected and zero-free graphical sequences of length n respectively.In Table 4 we show itemized potentially and forcibly biconnected graphical degree sequences of length 7 based on the degree sum N .The counts for N < 14 are not shown because those counts are all 0. The minimum N such that C 2 (7, N ) is nonzero can be easily determined to be 14 using the Wang and Kleitman [13] characterization.However, the minimum N such that F 2 (7, N ) is nonzero (which is equal to 20, not 14) is not obvious.In general for given n if the minimum N such that F 2 (n, N ) is nonzero can be easily determined, it can be added into Algorithm 2 so that for any input d with the sum of its terms less than this minimum the algorithm can immediately return False.The largest degree sum is 42 for any graphical degree sequence of length 7. Notice that both the nonzero C 2 (7, N ) values and the nonzero F 2 (7, N ) values when N increases form a unimodal sequence.Also notice that C 2 (7, N ) = F 2 (7, N ) when 28 ≤ N ≤ 42.This shows that for even N between 28 and 42, all potentially biconnected graphical partitions of N with 7 parts are also forcibly biconnected.That is, 28 is the smallest N such that all potentially biconnected graphical partitions of N with 7 parts are also forcibly biconnected.In Table 5 we show itemized numbers of forcibly biconnected graphical degree sequences of length 15 based on the largest degree.The counts for largest degrees less than 7 are not shown because those counts are all 0. Any graphical degree sequence of length 15 has a largest term at most 14.From the table we can see that the counts decrease when the largest degree decreases.For other degree sequence lengths from 5 to 25 we observed similar behavior.The table also indicates that there are no forcibly biconnected graphical degree sequences of length 15 with largest degree less than 7.In fact, we can define M 2 (n) to be the minimum largest term in any forcibly biconnected graphical sequence of length n.This is, M 2 (n) .= min{∆: ∆ is the largest term of some forcibly biconnected graphical degree sequence of length n}.Clearly we have M 2 (n) ≤ n/2 since for even n the sequence n/2, n/2, • • • , n/2 of length n is forcibly biconnected.In [14] we have defined M (n) to be the minimum largest term in any forcibly connected graphical sequence of length n (M (n) = M 1 (n) based on the notation M k (n) in Table 2).By definition we obviously have M 2 (n) ≥ M (n) so that we also have M 2 (n) > c √ n for all sufficiently large n and some constant c > 0 based on a lower bound of M (n) shown in [14].If M 2 (n) can be easily calculated, it can be added into Algorithm 2 so that any input d of length n with largest term less than M 2 (n) can be immediately decided to be non forcibly biconnected.We show the values of M 2 (n) and M (n) based on our enumerative results in Table 6.
We also incorporated our Algorithm 2 into the highly efficient Constant Amortized Time (CAT) algorithm of Barnes and Savage [1] to generate forcibly biconnected graphical partitions of even n.The results for n up to 170 together with the proportion of them in all forcibly connected graphical partitions of n are listed in Table 7.For the purpose of saving space we only show the results in increments of 10 for n.From the table it seems reasonable to conclude that the proportion g 2 f (n)/g 1 f (n) will decrease when n is beyond some small threshold and it might tend to the limit 0. Previously in [14] we have conjectured that lim n→∞ When generating all forcibly biconnected graphical partitions of n we can also output the itemized counts based on the number of parts or the largest part.In Table 8 we show itemized counts of potentially biconnected and forcibly biconnected graphical partitions of 30 based on the number of parts.The counts c 2 (30, j) and f 2 (30, j) for which the number of parts j is less than 6 or greater than 15 are not shown because those counts are all 0. When n is large, the minimum number of parts j for which c 2 (n, j) and f 2 (n, j) are both nonzero is clearly the smallest positive integer j(n) such that j(n)(j(n) − 1) ≥ n.The largest number of parts j for which c 2 (n, j) is nonzero is clearly n/2 since the sequence 2, • • • , 2 (n/2 copies) is potentially biconnected.The largest number of parts j for which f 2 (n, j) is nonzero does not appear to be easily computable.Note that the nonzero values of c 2 (30, j) and f 2 (30, j) both form a unimodal sequence when j increases.Also note that c 2 (30, j) = f 2 (30, j) for j = 6, 7.This shows that all potentially biconnected graphical partitions of 30 with 6 or 7 parts are also forcibly biconnected.That is, 7 is the largest number of parts j such that all potentially biconnected graphical partitions of 30 with j parts are also forcibly biconnected.
The fact that c 2 (30, 7) = f 2 (30, 7) = 16 agrees with the result that C 2 (7, 30) = F 2 (7, 30) = 16 in Table 4.In Table 9 we show itemized counts of forcibly biconnected graphical partitions of 30 based on the largest term.The counts l 2 (30, j) for which the largest part j is less than 4 or greater than 8 are not shown since those counts are all 0. The nonzero counts l 2 (30, j) form a unimodal sequence.Given n the largest j for which l 2 (n, j) is nonzero can be easily determined by solving the inequality n ≥ 2(j + 1) − 4 + 2j using the Wang and Kleitman [13] characterization since a potentially biconnected graphical degree sequence of length j + 1 with largest term j is forcibly biconnected.However, the smallest j for which l 2 (n, j) is nonzero does not appear to be easily computable.If we use m 2 (n) to denote this smallest j, i.e. the minimum largest term of any forcibly biconnected graphical partitions of n, then we clearly have 2 ≤ m 2 (n) ≤ √ n.We show the values of m 2 (n) for some small n based on our enumerative results in Table 10.The fact that m 2 (30) = 4 agrees with the fact that in Table 9 the counts l 2 (30, j) are all 0 for largest part j less than 4. In [14] we have defined m(n) as the minimum largest term of any forcibly connected graphical partition of n (m(n) = m 1 (n) in the notation m k (n) of Table 2).Clearly we have m(n) ≤ m 2 (n) by definition.We can see that m 2 (n) happen to agree with m(n) for those n listed in Table 10.4.3 Disproof of a previous conjecture regarding D 1 f (n) In [14] we conjectured that lim n→∞

Questions and conjectures
Based on the obtained enumerative results we ask the following questions and make certain conjectures: 1. What is the growth order of D 2 f (n) relative to D 1 f (n) and D(n)?We conjecture that both 2. What is the growth order of g 2 f (2n) relative to g 1 f (2n) and g(2n)?We conjecture that lim n→∞ g 2 f (2n) g 1 f (2n) = 0 and lim n→∞ g 2 f (2n) g(2n) = 0. Furthermore, we conjecture that g 2 f (2n)/g 1 f (2n) is monotonically decreasing when n ≥ 5. Is it true that lim n→∞ g k f (2n) g k−1 f (2n) = 0 for fixed k ≥ 3?
3. We conjecture that the numbers F 2 (n, N ) of forcibly biconnected graphical degree sequences of length n with degree sum N , when N runs through 2n − 2, 2n, • • • , n(n − 1), give a unimodal sequence.(There may be some zeros in the sequence.)4. Let t(n) be the smallest positive integer such that t(n)(t(n) − 1) ≥ n.We conjecture that the numbers f 2 (n, j) of forcibly biconnected graphical partitions of n with j parts, when j runs through t(n), t(n) + 1, • • • , n/2, give a unimodal sequence.(There may be some zeros in the sequence.)

Figure 1 :
Figure 1: A realization G of d with a cut vertex v c .
Remove a copy of d from d L if it appears at least once in d L ; if d L has at least s elements then 18 Form all possible s combinations s L of d L , add an element d with 1 ≤ d < d into s L to form a sequence s L of length s + 1; Form all possible sequences s H of length n − s − 1 consisting of the elements of d − ({d} ∪ s L ), add an element d = d − d (1 ≤ d < d) into s H to form a sequence s H of length n − s; If both s H and s L have even sum and are graphical, return False; return True; deg(v x ) is arbitrary.(G − {v c } is the disjoint union of G 1 and G 2 .See Figure 1.
then we need to ensure that d 2 , which is the largest degree d * in d other than d, does not exceed n − s − 1, i.e. d 2 ≤ n − s − 1.This inequality has already been satisfied due to the conditional test on line 7.If d < d 1 , then we need to ensure that d 1 , which is the largest degree d * in d other than d, does not exceed n − s − 1, i.e. d 1 ≤ n − s − 1.This completes our justification of the conditional test on line 13.

2 2
The number of choices of d S after d i is chosen depends on the magnitude of d i .Assume the average size of d i is n/2 for the purpose of simplicity of our analysis.The number of choices of d S is then about n−1 n/2 since d S is to be chosen from d − {d i }.Once d i and d S have been chosen, the sequence d of length n − 1 can be constructed and tested whether it is a graphical sequence in O(n) time.After that the testing of forcibly connectedness of d can be done in poly(n − 1) n−1(n−1)/2time.Thus we can see that the worst case run time complexity of the for loop from lines 3 to 6 is roughly poly(n) n n/under this simplified analysis.The total run time complexity of Algorithm 1 is dominated by the for loop from lines 3 to 6, which is about poly(n) n n/2 2 .

D 1 f
(n) D(n) = 1 purely based on the available enumerative results of D 1 f (n) for n up to 26.Now we refute this conjecture:Proposition 4.1.lim n→∞ D 1 f (n) D(n) = 1.Proof.Let D pd (n) denote the number of zero-free graphical degree sequences of length n that are potentially disconnected.ObviouslyD(n) = D 1 f (n) + D pd (n).We also have D pd (n) ≥ D(n − 2)since the set of zero-free potentially disconnected graphical degree sequences of length n includes those that are concatenations of a zero-free graphical degree sequence of length n−2 and the graphical degree sequence (1,1) of length 2. It is known from[15, Section  7]  that lim n→∞D(n−2) D(n) = 0. Now suppose lim n→∞ D 1 f (n) D(n) = 1.Then lim n→∞ D pd (n) D(n) = 0.This leads to lim n→∞ D(n−2) D(n) = 0, which is a contradiction.

D 2 f
(n) D(n) tend to a constant less than 1.However, they cannot tend to the same constant because lim n→∞D 1 f (n) D(n) = 1 according to Proposition 4.1.Is it true that lim n→∞ D 2 f (n) D 2 f (n−1) = 4? Furthermore, we conjecture that D 2 f (n) D 1 f (n) and D 2 f (n) D(n) areboth monotonically increasing when n ≥ 8. What can be said about the relative orders of D k c (n), D k f (n) and D(n) when k ≥ 3?
True or False, indicating whether d is forcibly biconnected or not 1 if d is not potentially biconnected or forcibly connected then 2 return False 3 for each d i and each collection d S of size d i from d − {d i } do 4 d ← GHH(d, d i , d S ); then consider two vertices v 2 and v x in the two disjoint induced subgraphs G 1 and G 2 of G separated by v c respectively such that deg(v 2 ) = d 2 and Algorithm 2: improved version pseudo-code to test whether a graphical degree sequence is forcibly biconnected Input: A zero-free graphical degree sequence d = (d 1 ≥ d 2 ≥ • • • ≥ d n ) Output: True or False, indicating whether d is forcibly biconnected or not 1 if d is not potentially biconnected or forcibly connected then then consider two vertices v 1 and v x in the two induced subgraphs G 1 and G 2 of G separated by v c respectively such that deg(v 1 ) = d 1 and deg(v x ) is arbitrary.By the same argument as above we have deg(v 1 )+deg(v x )−1 ≤ n − 2. Thus d 1 + deg(v x )≤ n − 1.Then we have d 2 + d n ≤ n − 1 since d 2 ≤ d 1 and d n ≤ deg(v x ).
[8]rees of the vertices of G 1 in G together with the degree d = d − d equal to the number of adjacencies of the cut vertex with G 1 .Clearly, one level of exhaustive enumeration comes from selecting s degrees s L from d L .The other level of exhaustive enumeration comes from choosing d (1 ≤ d < d), which is the number of adjacencies of the cut vertex with G 2 .Notationally we have s L = s L ∪ {d }.Once s L from d L and d (1 ≤ d < d) have been chosen, s L and s H are both determined, with s H = (d − ({d} ∪ s L )) ∪ {d }.We only need a linear time algorithm such as from[8]to test whether s L and s H are both graphical.If during the double for loop from lines 11 to 18 such a pair of graphical degree sequences s L and s H can be found, then we know that the input d is not forcibly biconnected (hence returning False on line 18) since we have found a cut degree d together with the degrees for vertices of G 1 and G 2 after cut and the number of adjacencies (d , d ) of the cut vertex with G 1 and G 2 respectively.If no such pair of graphical degree sequences s L and s H can ever be found on line 18, then we know that the input d is forcibly biconnected and we should return True on line 19.

Table 1 :
Chosen p l and p h in the experimental performance evaluation of Algorithm 2 for input length n = 500.

Table 2 :
Terminology used in Section 4.2

Table 5 :
Number L 2 (15, j) of forcibly biconnected graphical degree sequences of length 15 with given largest term j.

Table 7 :
Number of forcibly biconnected graphical partitions of n and their proportions in all forcibly connected graphical partitions of n.

Table 9 :
Number l 2 (30, j) of forcibly biconnected graphical partitions of 30 with given largest term j.