关于元爪图的一个性质(英文)

In this paper we consider a property of claw-free graphs.We show that if d（u）＋ d（v）≥ν（G）＋2k＋3,for every two nonadjacent vertices u and v,then G is 2k-vertex-deletable IM-extendable,whereν（G）=｜V（G）｜.And the bound is tight.     

INDEPENDENT-SET-DELETABLE FACTOR-CRITICAL POWER GRAPHS

It is said that a graph G is independent-set-deletable factor-critical (in short, ID-factor-critical), if, for every independent set 7 which has the same parity as |V(G)|, G-I has a perfect matching. A graph G is strongly IM-extendable, if for every spanning supergraph H of G, every induced matching of H is included in a perfect matching of H. The k-th power of G, denoted by Gk, is the graph with vertex set V(G) in which two vertices are adjacent if and only if they have distance at most k in G. ID-factor-criticality and IM-extendability of power graphs are discussed in this article. The author shows that, if G is a connected graph, then G3 and T(G) (the total graph of G) are ID-factor-critical, and G4 (when |V(G)| is even) is strongly IM-extendable; if G is 2-connected, then D2 is ID-factor-critical.     

无爪图的导出匹配可扩性

若图G的一个匹配M也是G的点导出子图,则称M是图G的一个导出匹配．我们称图G是导出匹配可扩的,若它的任何一个导出匹配可以扩充成一个完美匹配,本文我们讨论无爪图的导出匹配可扩性,得出如下结论,并同时指出这些结果是最好可能的．设图G是有2n个顶点的无爪图,1．若图G是最小度大于或等于2 1,则图G是导出匹配可扩的．2．若图G是局部2连通的,则留G是导出匹配可扩的．3．若图G是k正则的且k≥n,则图G是导出匹配可扩的．     

导出匹配,极大导出匹配可扩图和导出匹配数

目前我们已知的极大导出匹配可扩图只有Kn,n和K2n.为了研究它们是否是仅有的极大导出匹配可扩图,我们考虑了匹配数,导出匹配数,极大导出匹配可扩图以及一个相关的猜想,并得出了若干相关的结果.     

导出匹配问题的NP-完全性以及导出匹配可扩问题的CO-NP-完全性

图Ｇ的一个匹配Ｍ是导出的,若Ｍ是图Ｇ的一个导出子图。图Ｇ是导邮匹配可扩的（简记ＩＭ－可扩的）,若图Ｇ的任一导出匹配均含于图Ｇ的一个完美匹配当中。本文我们将证明如下结果。⑴对无爪图而言,问题“给定图Ｇ以及一个正整数ｒ,确定是否存在图Ｇ的一个导出匹配Ｍ使得Ｍ≥ｒ”是ＮＰ－完全的。⑵对直径为２的图以及直径为３的偶图,问题“确定一个给定图是否为导出匹配可扩的”是ＣＯ－ＮＰ完全的;而对完全多部图而言,问题“     

Induced Matching Extendable Graph Powers

A graph G is called induced matching extendable (shortly, IM-extendable) if every induced matching of G is included in a perfect matching of G. A graph G is called strongly IM-extendable if every spanning supergraph of G is IM-extendable. The k-th power of a graph G, denoted by G^k, is the graph with vertex set V(G) in which two vertices are adjacent if and only if the distance between them in G is at most k. We obtain the following two results which give positive answers to two conjectures of Yuan. Result 1. If a connected graph G with |V(G)| even is locally connected, then G² is strongly IM-extendable. Result 2. If G is a 2-connected graph with |V(G)| even, then G³ is strongly IM-extendable. Research Supported by NSFC Fund 10371102.     

Edge-deletable IM-extendable graphs with minimum number of edges

A graph G is induced matching extendable, shortly IM-extendable, if every induced matching of G is included in a perfect matching of G. For a nonnegative integer k, a graph G is called a k-edge-deletable IM-extendable graph, if, for every F⊆E(G) with |F|=k, G−F is IM-extendable. In this paper, we characterize the k-edge-deletable IM-extendable graphs with minimum number of edges. We show that, for a positive integer k, if G is ak-edge-deletable IM-extendable graph on 2n vertices, then |E(G)|≥(k+2)n; furthermore, the equality holds if and only if either G≅K_k+2,k+2, or k=4r−2 for some integer r≥3 and G≅C₅[N_2r], where N_2r is the empty graph on 2r vertices and C₅[N_2r] is the graph obtained from C₅ by replacing each vertex with a graph isomorphic to N_2r.