The matching-connectivity of a graph

Abstract

Let $H$ be a connected subgraph of a connected graph $G$. The $H$-structure connectivity of the graph $G$, denoted by $\kappa (G;H)$, is the minimum cardinality of a set of disjoint subgraphs $F=\{F_1,F_2,\dots ,F_m\}$ in $G$, such that every $F_i\in F$ is isomorphic to $H$ and $G-{\cup }_{F_i\in F}V\left(F_i\right)$ is disconnected or trivial. By definition, the vertex connectivity of a graph $G$ equals its $K_1$-structure connectivity, that is, $\kappa (G;K_1)=\kappa \left(G\right)$. Define ${\kappa }_M\left(G\right)=\kappa (G;K_2)$, known as the matching-connectivity of $G$.

In this paper, we prove that ${\kappa }_M\left(G\right)$ is well-defined if and only if $G\notin \{K_{2n},K_{n,n}\}$. For a connected graph $G\notin \{K_{2n},K_{n,n}\}$, we prove $\lceil \kappa \left(G\right)/2\rceil \le {\kappa }_M\left(G\right)\le \kappa \left(G\right)$. Moreover, we characterize the graphs $G$ satisfying ${\kappa }_M\left(G\right)=\lceil \kappa \left(G\right)/2\rceil$ for even $\kappa \left(G\right)$, as well as those satisfying ${\kappa }_M\left(G\right)=\kappa \left(G\right)$.