Graphs that are critical with respect to matching extension and diameter
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Let G be a simple connected graph on 2n vertices with a perfect matching. For 1 ≤ k ≤ n  1, G is said to be kextendable if for every matching M of size k in G there is a perfect matching in G containing all the edges of M. A kextendable graph G is said to be kcritical (kminimal) if G+uv (Guv) is not kextendable for every nonadjacent (adjacent) pair of vertices u and v of G. The problem that arises is that of characterizing kextendable, kcritical and kminimal graphs.In Chapter 2, we establish that δ(G) ≥ 1/2(n + k) is a sufficient condition for a bipartite graph G on 2n vertices to be kextendable. For a graph G on 2n vertices with δ(G) ≥ n + k 1, n  k even and n/2 ≤ k ≤ n  2, we prove that a necessary and sufficient condition for G to be kextendable is that its independence number is at most n  k. We also establish that a kextendable graph G of order 2n has k + 1 ≤ δ(G) n or δ(G) ≥ 2k + 1, 1 ≤ k ≤ n  1. Further, we establish the existence of a kextendable graph G on 2n vertices with δ(G) = j for each integer j Є [k + 1, n] u [2k + 1, 2n 1]. For k = n  1 and n  2, we completely characterize kextendable graphs on 2n vertices. We conclude Chapter 2 with a variation of the concept of extendability to odd order graphs.In Chapter 3, we establish a number of properties of kcritical graphs. These results include sufficient conditions for kextendable graphs to be kcritical. More specifically, we prove that for a kextendable graph G ≠ K2n on 2n vertices, 2 ≤ k ≤ n  1, if for every pair of nonadjacent vertices u and v of G there exists a dependent set S ( a subset S of V (G) is dependent if the induced subgraph G[S] has at least one edge) of Guv such that o(G(S u {u,v})) = S, then G is kcritical. Moreover, for k = 2 this sufficient condition is also a necessary condition for nonbipartite graphs. We also establish a necessary condition, in terms of the minimum degree, for kcritical graphs.We conclude Chapter 3 by completely characterizing kcritical graphs on 2n vertices for k = 1, n  1 and n  2.Chapter 4 contains results on kminimal graphs. These results include necessary and sufficient conditions for kextendable graphs to be kminimal. More specifically, we prove that for a kextendable graph G on 2n vertices, 1 ≤ k ≤ n  1, the following are equivalent:G is minimalfor every edge e = uv of G there exists a matching M of size k in Ge such that V(M) n {u,v} = ø and for every perfect matching F in G containing M, e Є F.for every edge e = uv of G there exists a vertex set S of Guv such that: M(S) ≥ k; o(GeS) = S  2k + 2; and u and v belong to different odd components of GeS, where M(S) denotes a maximum matching in G[S].We also establish a necessary condition, in terms of minimum degree, for kminimal and kminimal bipartite graphs. In fact, we prove that a kminimal graph G ≠ K2n on 2n vertices, 1 ≤ k ≤ n  1, has minimum degree at most n + k  1. For a kminimal bipartite graph G ≠ Kn,n , 1 ≤ k ≤ n  3, we show that δ(G) < ½(n + k).Chapter 1 provides the notation, terminology, general concepts and the problems concerning extendability graphs and (k,t)critical graphs.
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