Mixed connectivity of Cartesian graph products and bundles

Abstract

Mixed connectivity is a generalization of vertex and edge connectivity. A graph is $(p,0)$-connected, $p>0$, if the graph remains connected after removal of any $p-1$ vertices. A graph is $(p,q)$-connected, $p\geq 0$, $q>0$, if it remains connected after removal of any $p$ vertices and any $q-1$ edges. Cartesian graph bundles are graphs that generalize both covering graphs and Cartesian graph products. It is shown that if graph $F$ is $(p_{F},q_{F})$-connected and graph $B$ is $(p_{B},q_{B})$-connected, then Cartesian graph bundle $G$ with fibre $F$ over the base graph $B$ is $(p_{F}+p_{B},q_{F}+q_{B})$-connected. Furthermore, if $q_{F},q_{B}>0$, then $G$ is also $(p_{F}+p_{B}+1,q_{F}+q_{B}-1)$-connected. Finally, let graphs $G_i, i=1,...,n,$ be $(p_i,q_i)$-connected and let $k$ be the number of graphs with $q_i>0$. The Cartesian graph product $G=G_1\Box G_2\Box ... \Box G_n$ is $(\sum p_i,\sum q_i)$-connected, and, for $ k\geq 1$, it is also $(\sum p_i+k-1,\sum q_i-k+1)$-connected.