Total $k$-distance domination critical graphs

Abstract

A set $S$ of vertices in a graph $G=(V,E)$ is called a total‎ ‎$k$-distance dominating set if every vertex in $V$ is within‎ ‎distance $k$ of a vertex in $S$‎. ‎A graph $G$ is total $k$-distance‎ ‎domination-critical if $gamma_{t}^{k} (G‎ - ‎x) < gamma_{t}^{k}‎ ‎(G)$ for any vertex $xin V(G)$‎. ‎In this paper‎, ‎we investigate some results on total $k$-distance domination-critical of graphs‎.