直径为5的图的2-导出匹配划分和2-导出匹配覆盖问题的NP-完全性

给定一个简单图G和正整数κ,具有完美匹配的图G的κ-导出匹配划分是对顶点集V(C)的一个κ-划分(V1,V2,...,Vκ),其中对每一个i(1≤i≤κ),由Vi导出的G的子图G[Vi]是1-正则的.κ-导出匹配划分问题是指对给定的图G,判定G是否存在一个κ-导出匹配划分.令M1,M2…,Mκ为图G的κ个导出匹配,如果V(M1)UV(M2)∪...∪V(Mκ)=V(G),则我们称{M1,M2,...,Mκ}是G的κ-导出匹配覆盖.κ-导出匹配覆盖问题是指对给定的图G,判定G是否存在κ-导出匹配覆盖.本文给出了Yang,Yuan和Dong所提出问题的解,证明了直径为5的图的导出匹配2一划分问题和导出匹配2-覆盖问题都是NP-完全的.     

导出匹配可扩图的局部运算

称图G为导出匹配图可扩的（简称为IM－可扩的），如果图G的一每个导出匹配都包含在G的一个完美匹配中，本给出了导出匹配可扩图的一些局部运算。     

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.     

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．     

小直径图的导出匹配覆盖

设G是一个图,而M1,M2,…,Mk是G的k个导出匹配.称{M1,M2,…,Mk}是图G的一个k-导出匹配覆盖,若V(M1)∪V(M2)∪…∪V(Mk)=V(G).k-导出匹配覆盖问题是指对任一个给定的图G是否存在一个k-导出匹配覆盖.这篇文章证明了:直径为6的图的2-导出匹配覆盖问题和直径为2的图的3-导出匹配覆盖问题是NP-完备的,直径为2的图的2-导出匹配覆盖问题多项式可解.     

图的导出森林2-划分

对一个给定的简单图Ｇ，是否存在Ｖ（Ｇ）的一个２－划分（Ｖ１，Ｖ２）使得每个导出子图Ｇ［Ｖｉ］为森林？称该问题为导出森林２－划分问题．本文证明了对最大度为５的图该问题是ＮＰ－完全的，而对最大度≤４的图该问题多项式时间可解．     

图的路色数问题的NP－完全性

一个给定的图是否存在用ｒ种颜色的正常Ｐｋ着色？称该问题为图的（ｋ，ｒ）路色数问题．本文研究其算法复杂性，并得到以下结果：对于任意给定的ｋ，２≤ｋ≤∞，图的（ｋ，２）路色数问题及直径为２的图的（ｋ，３）路色数问题都是ＮＰ－完全的；对于任意给定的ｋ，２≤ｋ≤∞，平面图的（ｋ，３）路色数问题也是ＮＰ－完全的．     

偶匹配可扩性的极图问题(英文)

设G是含有完美匹配的简单图.称图G是偶匹配可扩的(BM-可扩的),如果G的每一个导出子图是偶图的匹配M都可以扩充为一个完美匹配.极图问题是图论的核心问题之一.本文将刻画极大偶匹配不可扩图,偶图图类和完全多部图图类中的极大偶匹配可扩图.     

直径为2的图的P2路色问题

一个给定的图是否存在用ｒ种颜色的正常Ｐｋ着色？称该问题为图的（ｋ，ｒ）路色数问题。已知对于直径为２的图及任意给定的整数ｒ≥３，图的（２，ｒ）路色数问题是ＮＰ－完全的。本文给出直径为２的（２，２）路色图的一个好的刻划，并由此给出该问题的一个多项式时间算法，从而解决了以ｒ为参数的直径为２的图的（２，ｒ）路色数问题的计算复杂性分类。     

偶匹配可扩性的极图问题

设G是含有完美匹配的简单图. 称图G是偶匹配可扩的(BM-可扩的), 如果G的每一个导出子图是偶图的匹配M都可以扩充为一个完美匹配. 极图问题是图论的核心问题之一. 本文将刻画极大偶匹配不可扩图, 偶图图类和完全多部图图类中的极大偶匹配可扩图.     

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.     

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.     

A Polynomial-time Algorithm for the Dominating Induced Matching Problem in the Class of Convex Graphs

The dominating induced matching problem is the problem of determining whether a graph has an induced matching that dominates every edge of the graph. This is known to be NP-complete in general. We develop a polynomial-time algorithm to solve the problem for convex graphs.     

Induced Matching Number of the Plane Grid Graph

An induced matching M in a graph G is a matching such that V(M) induces a 1-regular subgraph of G. The induced matching number of a graph G, denoted by I M(G), is the maximum number r such that G has an induced matching of r edges. Induced matching number of Pm×Pn is investigated in this paper. The main results are as follows:(1) If at least one of m and n is even, then IM(Pm×Pn=[(mn)/4].(2) If m is odd, then     

Induced matching extendable graphs

We say that a simple graph G is induced matching extendable, shortly IM-extendable, if every induced matching of G is included in a perfect matching of G. The main results of this paper are as follows: (1) For every connected IM-extendable graph G with |V(G)| ≥ 4, the girth g(G) ≤ 4. (2) If G is a connected IM-extendable graph, then |E(G)| ≥ ${3\over 2}|V(G)| - 2$; the equality holds if and only if G ≅ T × K₂, where T is a tree. (3) The only 3-regular connected IM-extendable graphs are C_n × K₂, for n ≥ 3, and C_2n(1, n), for n ≥ 2, where C_2n(1, n) is the graph with 2n vertices x₀, x₁, …, x_2n−1, such that x_ix_j is an edge of C_2n(1, n) if either |i − j| ≡ 1 (mod 2n) or |i − j| ≡ n (mod 2n). © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 203–213, 1998