A disjunctive total dominating set of a graph G is a set <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$D \subseteq V(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>⊆</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> such that every vertex in V(G) has a neighbor in D or has at least two vertices in D at distance 2 from it. The disjunctive total domination number of G, denoted by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\gamma _t^d(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>γ</mi> <mi>t</mi> <mi>d</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, is the minimum cardinality among all disjunctive total dominating sets of G. In this paper, we show that if G is a maximal outerplanar graph with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n\ge 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> vertices, then <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\gamma _t^d(G)\le \lfloor \frac{1}{3}(n+1)\rfloor $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>γ</mi> <mi>t</mi> <mi>d</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mrow> <mo>⌊</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>⌋</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and this bound is sharp.

Disjunctive total domination in maximal outerplanar graphs

期刊论文
发表日期：

2026年5月29日

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Michael A. Henning,
Paras Vinubhai Maniya,
Dinabandhu Pradhan

摘要

A disjunctive total dominating set of a graph G is a set $D \subseteq V(G)$ $D \subseteq V (G)$ such that every vertex in V(G) has a neighbor in D or has at least two vertices in D at distance 2 from it. The disjunctive total domination number of G, denoted by $\gamma _t^d(G)$ $γ_{t}^{d} (G)$ , is the minimum cardinality among all disjunctive total dominating sets of G. In this paper, we show that if G is a maximal outerplanar graph with $n\ge 5$ $n \geq 5$ vertices, then $\gamma _t^d(G)\le \lfloor \frac{1}{3}(n+1)\rfloor $ $γ_{t}^{d} (G) \leq ⌊ \frac{1}{3} (n + 1) ⌋$ , and this bound is sharp.