Let G be a simple graph and T={S :S is extreme in G}. If M(V(G), T) is a matroid, then G is called an extreme matroid graph. In this paper, we study the properties of extreme matroid graph.
For a graph G =(V,E),a subset VS is a dominating set if every vertex in V is either in S or is adjacent to a vertex in S.The domination number γ(G) of G is the minimum order of a dominating set in G.A graph G is said...For a graph G =(V,E),a subset VS is a dominating set if every vertex in V is either in S or is adjacent to a vertex in S.The domination number γ(G) of G is the minimum order of a dominating set in G.A graph G is said to be domination vertex critical,if γ(G-v) < γ(G) for any vertex v in G.A graph G is domination edge critical,if γ(G ∪ e) < γ(G) for any edge e ∈/E(G).We call a graph G k-γ-vertex-critical(resp.k-γ-edge-critical) if it is domination vertex critical(resp.domination edge critical) and γ(G) = k.Ananchuen and Plummer posed the conjecture:Let G be a k-connected graph with the minimum degree at least k+1,where k 2 and k≡|V|(mod 2).If G is 3-γ-edge-critical and claw-free,then G is k-factor-critical.In this paper we present a proof to this conjecture,and we also discuss the properties such as connectivity and bicriticality in 3-γ-vertex-critical claw-free graph.展开更多
The perfect matching polytope of a graph G is the convex hull of the incidence vectors of all perfect matchings of G.A graph G is PM-compact if the 1-skeleton graph of the prefect matching polytope of G is complete.Eq...The perfect matching polytope of a graph G is the convex hull of the incidence vectors of all perfect matchings of G.A graph G is PM-compact if the 1-skeleton graph of the prefect matching polytope of G is complete.Equivalently,a matchable graph G is PM-compact if and only if for each even cycle C of G,G-V(C)has at most one perfect matching.This paper considers the class of graphs from which deleting any two adjacent vertices or nonadjacent vertices,respectively,the resulting graph has a unique perfect matching.The PM-compact graphs in this class of graphs are presented.展开更多
文摘Let G be a simple graph and T={S :S is extreme in G}. If M(V(G), T) is a matroid, then G is called an extreme matroid graph. In this paper, we study the properties of extreme matroid graph.
基金supported by Major State Basic Research Development Program of China (973 Project) (Grant No.2006CB805904)Natural Sciences and Engineering Research Council of Canada (Grant No.122059-200)
文摘For a graph G =(V,E),a subset VS is a dominating set if every vertex in V is either in S or is adjacent to a vertex in S.The domination number γ(G) of G is the minimum order of a dominating set in G.A graph G is said to be domination vertex critical,if γ(G-v) < γ(G) for any vertex v in G.A graph G is domination edge critical,if γ(G ∪ e) < γ(G) for any edge e ∈/E(G).We call a graph G k-γ-vertex-critical(resp.k-γ-edge-critical) if it is domination vertex critical(resp.domination edge critical) and γ(G) = k.Ananchuen and Plummer posed the conjecture:Let G be a k-connected graph with the minimum degree at least k+1,where k 2 and k≡|V|(mod 2).If G is 3-γ-edge-critical and claw-free,then G is k-factor-critical.In this paper we present a proof to this conjecture,and we also discuss the properties such as connectivity and bicriticality in 3-γ-vertex-critical claw-free graph.
基金supported by the National Natural Science Foundation of China(Nos.12171440,11971445)。
文摘The perfect matching polytope of a graph G is the convex hull of the incidence vectors of all perfect matchings of G.A graph G is PM-compact if the 1-skeleton graph of the prefect matching polytope of G is complete.Equivalently,a matchable graph G is PM-compact if and only if for each even cycle C of G,G-V(C)has at most one perfect matching.This paper considers the class of graphs from which deleting any two adjacent vertices or nonadjacent vertices,respectively,the resulting graph has a unique perfect matching.The PM-compact graphs in this class of graphs are presented.
