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

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 $$\gamma _{pr}(G)$$ γ p r ( G ) . Let G be a connected $$\{K_{1,3}, K_{4}-e\}$$ { K 1 , 3 , K 4 - e } -free cubic graph of order n. We show that $$\gamma _{pr}(G)\le \frac{10n+6}{27}$$ γ p r ( G ) ≤ 10 n + 6 27 if G is $$C_{4}$$ C 4 -free and that $$\gamma _{pr}(G)\le \frac{n}{3}+\frac{n+6}{9(\lceil \frac{3}{4}(g_o+1)\rceil +1)}$$ γ p r ( G ) ≤ n 3 + n + 6 9 ( ⌈ 3 4 ( g o + 1 ) ⌉ + 1 ) if G is $$\{C_{4}, C_{6}, C_{10}, \ldots , C_{2g_o}\}$$ { C 4 , C 6 , C 10 , … , C 2 g o } -free for an odd integer $$g_o\ge 3$$ g o ≥ 3 ; the extremal graphs are characterized; we also show that if G is a 2 -connected, $$\gamma _{pr}(G) = \frac{n}{3} $$ γ p r ( G ) = n 3 . Furthermore, if G is a connected $$(2k+1)$$ ( 2 k + 1 ) -regular $$\{K_{1,3}, K_4-e\}$$ { K 1 , 3 , K 4 - e } -free graph of order n, then $$\gamma _{pr}(G)\le \frac{n}{k+1} $$ γ p r ( G ) ≤ n k + 1 , with equality if and only if $$G=L(F)$$ G = L ( F ) , where $$F\cong K_{1, 2k+2}$$ F ≅ K 1 , 2 k + 2 , or k is even and $$F\cong K_{k+1,k+2}$$ F ≅ K k + 1 , k + 2 .