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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.)
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[1] Duckworth, W; Wormald, NC, Minimum independent dominating sets of random cubic graphs, Electron. J. Comb., 9, 147-161, (2002) · Zbl 1009.05106
[2] Goddard, W; Henning, MA; McPillan, CA, Semitotal domination in graphs, Util. Math., 94, 67-81, (2014) · Zbl 1300.05220
[3] Haas, R; Seyffarth, K, The k-dominating graph, Graph Comb., 30, 609-617, (2014) · Zbl 1294.05122
[4] Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in Graphs: Advanced Topics. Marcel Dekker, New York (1998) · Zbl 0883.00011
[5] Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Marcel Dekker, New York (1998) · Zbl 0890.05002
[6] Henning, MA, A survey of selected recent results on total domination in graphs, Discrete Math., 309, 32-63, (2009) · Zbl 1219.05121
[7] Henning, MA; Löwnstein, C, Locating-total domination in claw-free cubic graphs, Discrete Math., 312, 3107-3116, (2012) · Zbl 1252.05162
[8] Henning, MA; Marcon, AJ, Semitotal domination in claw-free cubic graphs, Ann. Comb., 20, 1-15, (2016) · Zbl 1354.05107
[9] Henning, M.A., Yeo, A.: Total Domination in Graphs. Springer, New York (2013) · Zbl 1408.05002
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