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 $$\gamma _{pr}(G)$$źpr(G). Let G be a connected $$\{K_{1,3}, K_{4}-e\}$${K1,3,K4-e}-free cubic graph of order n. We show that $$\gamma _{pr}(G)\le \frac{10n+6}{27}$$źpr(G)≤10n+627 if G is $$C_{4}$$C4-free and that $$\gamma _{pr}(G)\le \frac{n}{3}+\frac{n+6}{9(\lceil \frac{3}{4}(g_o+1)\rceil +1)}$$źpr(G)≤n3+n+69(ź34(go+1)ź+1) if G is $$\{C_{4}, C_{6}, C_{10}, \ldots , C_{2g_o}\}$${C4,C6,C10,ź,C2go}-free for an odd integer $$g_o\ge 3$$goź3; the extremal graphs are characterized; we also show that if G is a 2 -connected, $$\gamma _{pr}(G) = \frac{n}{3} $$źpr(G)=n3. Furthermore, if G is a connected $$(2k+1)$$(2k+1)-regular $$\{K_{1,3}, K_4-e\}$${K1,3,K4-e}-free graph of order n, then $$\gamma _{pr}(G)\le \frac{n}{k+1} $$źpr(G)≤nk+1, with equality if and only if $$G=L(F)$$G=L(F), where $$F\cong K_{1, 2k+2}$$FźK1,2k+2, or k is even and $$F\cong K_{k+1,k+2}$$FźKk+1,k+2.