Paired-domination number of claw-free odd-regular graphs

Abstract

A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching, while the paired-domination number is the minimum cardinality of a paired-dominating set in the graph, denoted by $\gamma _{pr}(G)$. Let G be a connected $\{K_{1,3}, K_{4}-e\}$-free cubic graph of order n. We show that $\gamma _{pr}(G)\le \frac{10n+6}{27}$ if G is $C_{4}$-free and that $\gamma _{pr}(G)\le \frac{n}{3}+\frac{n+6}{9(\lceil \frac{3}{4}(g_o+1)\rceil +1)}$ if G is $\{C_{4}, C_{6}, C_{10}, \ldots , C_{2g_o}\}$-free for an odd integer $g_o\ge 3$; the extremal graphs are characterized; we also show that if G is a 2 -connected, $\gamma _{pr}(G) = \frac{n}{3}$. Furthermore, if G is a connected $(2k+1)$-regular $\{K_{1,3}, K_4-e\}$-free graph of order n, then $\gamma _{pr}(G)\le \frac{n}{k+1}$, with equality if and only if $G=L(F)$, where $F\cong K_{1, 2k+2}$, or k is even and $F\cong K_{k+1,k+2}$.