EDGE WEIGHTING FUNCTIONS ON THE SEMITOTAL DOMINATING SET OF CLAW-FREE GRAPHS.

CHEN, JIE; CHEN, HONGZHANG; XU, SHOU-JUN

In an isolate-free graph G , a subset S of vertices is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number of G , denoted by $\gamma _{t2}(G)$ , is the minimum cardinality of a semitotal dominating set in G. Using edge weighting functions on semitotal dominating sets, we prove that if $G\neq N_2$ is a connected claw-free graph of order $n\geq 6$ with minimum degree $\delta (G)\geq 3$ , then $\gamma _{t2}(G)\leq \frac{4}{11}n$ and this bound is sharp, disproving the conjecture proposed by Zhu et al. ['Semitotal domination in claw-free cubic graphs', Graphs Combin. 33 (5) (2017), 1119–1130].

GRAPH connectivity; DOMINATING set; LOGICAL prediction

Bulletin of the Australian Mathematical Society, 2024, Vol 110, Issue 2, p177

0004-9727

Academic Journal

10.1017/S0004972724000017