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Semitotal domination in claw-free cubic graphs. (English) Zbl 1386.05146
W. Goddard et al. [Util. Math. 94, 67–81 (2014; Zbl 1300.05220)] introduced semitotal domination in graphs. Let $$V(G)$$ be the vertex set of a graph $$G$$ containing no isolated vertices. A set $$S \subseteq V(G)$$ is called a semitotal dominating set of $$G$$ if each vertex not in $$S$$ is adjacent to a vertex in $$S$$, and each vertex in $$S$$ is at distance at most two from another vertex of $$S$$. The semitotal domination number of $$G$$, denoted by $$\gamma_{t2}(G)$$, is the minimum cardinality over all semitotal dominating sets of $$G$$.
For $$i\in\{1,2\}$$, let $$D_i$$ be a complete graph of order $$4$$ minus one edge; let $$V(D_i)=\{a_i, b_i, c_i, d_i\}$$ and let $$a_ib_i$$ be the missing edge in $$D_i$$. We denote by $$N_2$$ the graph obtained from the disjoint union of $$D_1$$ and $$D_2$$ by adding two edges $$a_1b_2$$ and $$a_2b_1$$, and we denote by $$K_4$$ the complete graph on $$4$$ vertices.
M. A. Henning and A. J. Marcon [Ann. Comb. 20, No. 4, 799–813 (2016; Zbl 1354.05107)] showed that, if $$G$$ is a connected claw-free cubic graph of order $$n \geq 10$$, then $$\gamma_{t2}(G) \leq \frac{4n}{11}$$, and they posed the following conjecture.
(C1) If $$G \not\in \{K_4, N_2\}$$ is a connected claw-free cubic graph of order $$n$$, then $$\gamma_{t2}(G) \leq \frac{n}{3}$$.
In this paper, the authors prove the above conjecture (C1), and they propose the following conjecture.
(C2) If $$G \not\in \{K_4, N_2\}$$ is a connected claw-free graph of order $$n$$ with minimum degree at least $$3$$, then $$\gamma_{t2}(G) \leq \frac{n}{3}$$.

##### MSC:
 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
##### Keywords:
semitotal domination; cubic graph; claw-free graph
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##### References:
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