Reducing the maximum degree of a graph by deleting vertices

We investigate the smallest number λ(G) of vertices that need to be removed from a non-empty graph G so that the resulting graph has a smaller maximum degree. We prove that if n is the number of vertices, k is the maximum degree, and t is the number of vertices of degree k, then λ(G) ≤ n+(k−1)t 2k . We also show that λ(G) ≤ n k+1 if G is a tree. These bounds are sharp. We provide other bounds together with structural observations.


Introduction
Throughout this paper we shall use capital letters such as X to denote sets or graphs, and small letters such as x to denote non-negative integers or elements of a set. The set {1, 2, . . . } of positive integers is denoted by N. For any n ∈ N, the set {1, . . . , n} is denoted by [n]. For a set X, the set {{x, y} : x, y ∈ X, x = y} of all 2-element subsets of X is denoted by X 2 . It is to be assumed that arbitrary sets are finite. A graph G is a pair (X, Y ), where X is a set, called the vertex set of G, and Y is a subset of X 2 and is called the edge set of G. The vertex set of G and the edge set of G are denoted by V (G) and E(G), respectively. It is to be assumed that arbitrary graphs have non-empty vertex sets. An element of V (G) is called a vertex of G, and an element of E(G) is called an edge of G. We may represent an edge {v, w} by vw. If vw is an edge of G, then v and w are said to be adjacent in G, and we say that w is a neighbour of v in G (and vice-versa). An edge vw is said to be incident to x if x = v or x = w.
For any v ∈ V (G), N G (v) denotes the set of neighbours of v in G, N G [v] denotes N G (v) ∪ {v} and is called the closed neighbourhood of v in G, and d G (v) denotes |N G (v)| and is called the degree of v in G. For X ⊆ V (G), we denote v∈X N G (v) and v∈X N G [v] by N G (X) and N G [X], respectively. The minimum degree of G is min{d G (v) : v ∈ V (G)} and is denoted by δ(G). The maximum degree of G is max{d G (v) : v ∈ V (G)} and is denoted by Δ(G). If G = (∅, ∅), then we take both δ(G) and Δ(G) to be 0.
If H is a graph such that V (H) ⊆ V (G) and E(H) ⊆ E(G), then H is said to be a subgraph of G, and we say that G contains H. For X ⊆ V (G), (X, E(G) ∩ X 2 ) is called the subgraph of G induced by X and is denoted by G [X]. For a set S, G − S denotes the subgraph of G obtained by removing from G the vertices in S and all edges incident to them, that is, In this paper, we investigate the minimum number of vertices that need to be removed from a graph so that the new graph obtained has a smaller maximum degree.
Let M(G) denote the set of vertices of G of degree Δ(G). We call a subset R We provide several bounds for λ(G). Our main results are given in the next section. Before stating our results, we need further definitions and notation. For The domination number of G, denoted by γ(G), is the size of a smallest dominating set of G.
We now define some special graphs and important concepts. If n ≥ 2 and v 1 , v 2 , . . . , v n are the distinct vertices of a graph G with E(G) = {v i v i+1 : i ∈ [n − 1]}, then G is called a v 1 v n -path or simply a path. The path ([n], {{1, 2}, . . . , {n − 1, n}}) is denoted by P n . For a path P , the length of P , denoted by l(P ), is |V (P )| − 1 (the number of edges of P ).
For u, v ∈ V (G), the distance of v from u, denoted by d G (u, v), is given by one that is not a subgraph of any other connected subgraph of G). It is easy to see that if H and K are distinct components of a graph G, then H and K have no common vertices (and therefore no common edges). If G 1 , . . . , G r are the distinct components of G, then we say that G is the disjoint union of G 1 , . . . , G r .
If n ≥ 3 and v 1 , v 2 , . . . , v n are the distinct vertices of a graph G with E(G) A graph G is a tree if G is a connected graph that contains no cycles. A graph G is a star if E(G) = {uv : v ∈ V (G)\{u}} for some u ∈ V (G). Thus a star is a tree.
A graph G is complete if every two vertices of G are adjacent (that is, ). A graph G is empty if no two vertices of G are adjacent (that is, E(G) = ∅). A graph G is regular if the degrees of its vertices are the same. If k ∈ {0} ∪ N and the degree of each vertex of G is k, then G is called k-regular.
Let H be a graph. A graph G is a copy of H if there exists a bijection f : We are now ready to state our main results, given in the next section. In Section 3, we investigate λ(G) from a structural point of view, particularly observing how this parameter changes with the removal of vertices. Some of the structural results are then used in the proofs of the main results; these proofs are given in Section 4.

Bounds
Our first result is a lower bound for λ(G). The bound above is sharp; for example, it is attained by complete graphs. We now provide a number of upper bounds for λ(G).

Proposition 2.2
For any non-empty graph G, Let d(G) denote the average degree 1 It immediately follows that λ(G) ≤ 1 2 |V (G)|. In Section 4, we characterize the cases in which the bound 1 2 |V (G)| is attained.

Theorem 2.3
For any non-empty graph G, and equality holds if and only if G is either a disjoint union of copies of K 2 or a disjoint union of copies of C 4 .
The subsequent new theorems in this section are also proved in Section 4. The following sharp bound is our primary contribution.
We point out four facts regarding Theorem 2.4. The first is that it immediately implies (1).
. Secondly, the bound in Theorem 2.4 can be attained in cases where λ(G) = t and also in cases where λ(G) < t. If G is a disjoint union of t copies of K 1,k , then λ(G) = t, n = (k + 1)t, and hence λ( . If G is one of the extremal structures in Theorem 2.3, then t = n and λ(G) = n 2 = n+(k−1)t 2k . Thirdly, it is immediate from the proof of Theorem 2.4 that the inequality in the result is strict if the closed neighbourhood of some vertex of G contains at least 3 members of M(G); see (7).
and if n k+1 < n+(k−1)t 2k and k ≥ 2, then n < (k + 1)t and λ(G) It turns out that if G is a tree, then, although we may have n k+1 < n+(k−1)t 2k (that is, n < (k + 1)t, as in the case of trees that are paths with at least 4 vertices), λ(G) ≤ n k+1 holds.

Theorem 2.5 For any tree T ,
The bound is sharp; for example, it is attained by stars. By Proposition 2.2, any upper bound for γ(G) is an upper bound for λ(G). Domination is widely studied and several bounds are known for γ(G); see [4]. The following well-known domination bound of Reed [9] gives us λ(G) Arnautov [3], Payan [8] and Lovász [7] independently proved that Alon and Spencer [2] gave a probabilistic proof using Alon's well-known argument in [1]. By adapting the argument to our problem of dominating M(G) rather than all of V (G), we prove the following improved bound for λ(G), replacing in particular δ(G) by Δ(G).
We conclude this section with a brief discussion on regular graphs. If G is regular, then M(G) = V (G), and hence λ(G) = γ(G). For a regular graph G, Theorem 2.7 is given by (3) as δ(G) = Δ(G). Kostochka and Stodolsky [6] obtained an improvement of the bound in Theorem 2.6 for 3-regular graphs.
Also, they showed in [5] that there exists an infinite class of connected 3-regular . This means that the lower bound in Proposition 2.1 is not always attained by regular graphs, and that the bound in Theorem 2.5 does not extend to the class of regular graphs. For regular graphs G with Δ(G) ≤ 2, the problem is trivial. Indeed, if such a graph G is connected, then either G has only one edge or G is a cycle. It is easy to check that {1+3t : 1+3t ∈ [n]} is a Δ-reducing set of C n of minimum size, and hence λ(C n ) = n 3 = |V (Cn)| Δ(Cn)+1 .

Structural results
In this section, we provide some observations on how λ(G) is affected by the structure of G and by removing vertices or edges from G. Some of the following facts are used in the proofs of our main results.
The result follows. 2 We point out that having |R| = λ(G) in Lemma 3.1 does not guarantee that |R ∩ V (H)| = λ(H). Indeed, let k ≥ 2, let G 1 and G 2 be copies of K 1,k such that V (G 1 ) ∩ V (G 2 ) = ∅, let G be the disjoint union of G 1 and G 2 , let e be an edge of G is a graph and G 1 , . . . , G r are the distinct components of G whose maximum degree is Δ(G), then λ(G) = r i=1 λ(G i ).

Proposition 3.2 If
Proof. Let R be a Δ-reducing set of G of size λ(G), and let Thus R is a Δ-reducing set of G, and hence λ( v is a vertex of a graph G, then λ(G) ≤ 1 + λ(G − v).

Proofs of the main results
We now prove Theorems 2.3, 2.4, 2.5 and 2.7.
Proof of Theorem 2.3. Let n = |V (G)| and k = Δ(G). Since G is non-empty, k > 0. By (1), λ(G) ≤ n 2 . It is straightforward that if G is either a disjoint union of copies of K 2 , or a disjoint union of copies of C 4 , then λ(G) = n 2 . We now prove the converse. Thus, suppose λ(G) = n 2 . Then, by (1), G is k-regular. Let G 1 , . . . , G r be the distinct components of G. Consider any i ∈ [r].
Applying the established bound to each of G 1 , . . . , G r , we have λ for each j ∈ [r]. Together with Proposition 3.2, this gives us r j=1 Therefore, k ≤ 2. If k = 1, then G i is a copy of K 2 . Suppose k = 2. Clearly, a 2-regular graph can only be a cycle. Thus, for some p ≥ 3, G i is a copy of C p . As pointed out in Section 2, λ(

Proof of Theorem 2.4.
Since G is non-empty, k > 0. Let r = λ(G) and G 1 = G. Let R be a Δ-reducing set of G of size r. We remove from G 1 a vertex v 1 in R whose closed neighbourhood in G 1 contains the largest number of vertices in M(G 1 ), and we denote the resulting graph G 1 − v 1 by G 2 . If r ≥ 2, then we remove from G 2 a vertex v 2 in R\{v 1 } whose closed neighbourhood in G 2 contains the largest number of vertices in M(G 2 ), and we denote the resulting graph G 2 − v 2 by G 3 . If r ≥ 3, then we remove from G 3 a vertex v 3 in R\{v 1 , v 2 } whose closed neighbourhood in G 3 contains the largest number of vertices in M(G 3 ), and we denote the resulting graph Continuing this way, we obtain v 1 , . . . , v r and G 1 , . . . , G r+1 such that The members v 1 , . . . , v r of R have been labelled in such a way that

and hence
Let  (4), M(G) = i∈I 2 ∪I 3 A i . By (6), it follows that t = i∈I 2 ∪I 3 |A i | ≥ i∈I 2 ∪I 3 2 = 2r, and hence r ≤ Now suppose r 1 = 0. Then Δ(H) = k. By construction, {v i : i ∈ I 1 } is a Δreducing set of H, and M(H) = i∈I 1 A i . If we assume that H has a Δ-reducing set S of size less than |I 1 |, then we obtain that (R\{v i : i ∈ I 1 }) ∪ S is a Δ-reducing set of G of size less than |R|, a contradiction. Thus λ(H) = |I 1 |. Together with M(H) = i∈I 1 A i , (6) gives us |M(H)| = i∈I 1 |A i | = |I 1 |. By Proposition 3.6, M(H) = M 2 (H). For each i ∈ I 1 , let z i be the unique element of A i . By (6), , we obtain that (R \{v j , z i }) ∪ {w 2 } is a Δ-reducing set of G of size |R | − 1, which contradicts |R | = λ(G). Thus w 2 ∈ C\{v j }, meaning that w 2 = v i for some i ∈ I 2 such that i > j. From this we obtain that R \{v j } is a Δ-reducing set of G of size |R | − 1, a contradiction. Therefore, v j = w 1 . Similarly, v j = w 2 . If we assume that w 1 , w 2 ∈ C, then we obtain that R \{v j } is a Δ-reducing set of G of size |R | − 1, a contradiction. Therefore, at least one of w 1 and w 2 is in B 1 ; we may assume that w 1 ∈ B 1 . Thus w 1 ∈ N H [z i ] for some i ∈ I 1 . If we assume that w 2 ∈ C, then we obtain that R \{v j } is a Δ-reducing set of G of size |R | − 1, a contradiction. Thus w 2 ∈ B 1 , and hence w 2 ∈ N H [z h ] for some h ∈ I 1 . From this we obtain that R \{v j } is a Δ-reducing set of G of size |R | − 1, a contradiction. Therefore, By (4) and (6), the sets A 1 , . . . , A r partition M(G). Thus t = r i=1 |A i | ≥ 3r 3 + 2r 2 + r 1 = 2r 3 + r 2 + r, and hence −r 3 − r 2 ≥ r − t + r 3 . We have If k = 1, then r 3 = 0. Thus (k − 2)r 3 ≥ 0, and hence r ≤ n+(k−1)t 2k . 2 We now prove Theorem 2.5, making use of the following well-known fact.
If v is adjacent to a vertex w of distance i, then, by considering an xv-path and an xw-path, we obtain that T contains a cycle, which is a contradiction. We obtain the same contradiction if we assume that v is adjacent to two vertices of distance i − 1 from x. If a vertex v of a graph G has only one neighbour in G, then v is called a leaf of G. Then each component of T is a tree. Let K be the set of components of T whose maximum degree is k, and let H be the set of components of T whose maximum degree is less than k. Let W = {w, z 1 , . . . , z k−1 }. Note that (W, {wz 1 , . . . , wz k−1 }) is in H, and hence W ∩ K∈K V (K) = ∅. If K = ∅, then {v} is a Δ-reducing set of T , and hence λ(T ) = 1 ≤ n k+1 . Suppose K = ∅. For each K ∈ K, let S K be a Δ-reducing set of K of size λ(K). By the induction hypothesis, |S K | ≤ |V (K)| k+1 for each K ∈ K. Now {v} ∪ K∈K S K is a Δ-reducing set of T . Therefore, we have as required. 2 Proof of Theorem 2.7. We may assume that V (G) = [n]. Let p = ln(k+1) k+1 . We set up n independent random experiments, and in each experiment a vertex is chosen with probability p. More formally, for each i ∈ V , let (Ω i , P i ) be the probability space given by Ω i = {0, 1}, P i ({1}) = p and P i ({0}) = 1 − p. Let Ω = Ω 1 × · · · × Ω n , and let P : 2 Ω → [0, 1] such that P ({ω}) = n i=1 P i ({ω i }) for each ω = (ω 1 , . . . , ω n ) ∈ Ω, and P (A) = ω∈A P ({ω}) for each A ⊆ Ω. Then (Ω, P ) is a probability space.
For each ω = (ω 1 , . . . , ω n ) ∈ Ω, let S ω be the subset of V (G) such that ω is the characteristic vector of S ω (that is, S ω = {i ∈ [n] : ω i = 1}), let T ω be the set of vertices in M(G) that are neither in S ω nor adjacent to a vertex in S ω (that is, T ω = {v ∈ M(G) : v / ∈ N G [S ω ]}), and let D ω = S ω ∪ T ω . Then D ω is a Δ-reducing set of G.