Edge criticality in secure graph domination

A subset $X$ of the vertex set of a graph $G$ is a secure dominating set of $G$ if $X$ is a dominating set of $G$ and if, for each vertex $u$ not in $X$ , there is a neighbouring vertex $v$ of $u$ in $X$ such that the swap set $X‐\left\{v\right\}\cup \left\{u\right\}$ is again a dominating set of $G$ . The secure domination number of $G$ is the cardinality of a smallest secure dominating set of $G$ . A graph $G$ is $q$ ‐critical if the smallest arbitrary subset of edges whose removal from $G$ necessarily increases the secure domination number, has cardinality $q$ . In this paper we characterise $q$ ‐critical graphs for all admissible values of $q$ and determine the exact values of $q$ for which members of various infinite classes of graphs are $q$ ‐critical.