Total domination in plane triangulations

TL;DR

This work studies the total domination number in plane near-triangulations and proves that for any near-triangulation $G$ of order $n\ge 5$, $\gamma_t(G) \le floor(2n/5)$ except for two graphs $H1$ and $H2$. The proof extends results known for maximal outerplanar graphs by introducing reducible/irreducible near-triangulations and terminal polygons, and employing edge contractions and deletions to drive a tight induction. Compared with the previous $6/11$-type bound, this yields a substantially stronger upper bound and provides a constructive way to obtain total dominating sets, while explicitly handling the exceptional cases. The authors also conjecture a sharper bound for triangulations, $\gamma_t(T) \le floor(n/3)$, and highlight the terminal-polygon technique as a potentially broadly useful tool in domination problems on planar graphs.

Abstract

A total dominating set of a graph $G=(V,E)$ is a subset $D$ of $V$ such that every vertex in $V$ is adjacent to at least one vertex in $D$. The total domination number of $G$, denoted by $γ_t (G)$, is the minimum cardinality of a total dominating set of $G$. A near-triangulation is a biconnected planar graph that admits a plane embedding such that all of its faces are triangles except possibly the outer face. We show in this paper that $γ_t (G) \le \lfloor \frac{2n}{5}\rfloor$ for any near-triangulation $G$ of order $n\ge 5$, with two exceptions.

Figures (8)

• Figure 1: The two 12-vertex graphs $H_1$ and $H_2$
• Figure 2: (a) A near-triangulation. The thick segments correspond to $T[C]$. (b) Removing a vertex of degree 2 in $T[C]$. (c) Contracting the edge $(u_i,u_{i+1})$ to the vertex $u'_i$.
• Figure 3: (a) A near-triangulation $T$ without diagonals. (b), (c), (d) and (e) Obtaining the near-triangulations $T_2, T_3, T_4$ and $T_5$ by removing successively the vertices $u_2, u_3, u_4$ and $v_4$.
• Figure 4: Illustrating Lemma \ref{['lem:ExceptionCases']}. In each case, the squared vertices form a TDS of $T$. (a) The graph $H_1$. (b) Removing the vertex $u_i$. (c) Removing the vertices $u_i$ and $v_i$. (d) The vertex $w_3$ is the vertex obtained by contracting the edge $(u_i,v_i)$. (e) and (f) Removing the edge $(u_i, u_{i+1})$.
• Figure 5: (a) The simplest irreducible near-triangulation $H$. The squared vertices form a total dominating set. (b) A irreducible near-triangulation $T$. Thick lines correspond to the subgraph $T[C]$. The diagonals of $T[C]$ define a set of adjacent polygons, five of which are non-empty and three are terminal, $P_2,P_3$ and $P_5$.

Theorems & Definitions (23)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• proof
• Claim 1
• Lemma 5
• proof
• Lemma 6: Dorfling16Lemanska17
• Lemma 7: Dorfling16Lemanska17