On three domination-based identification problems in block graphs

TL;DR

This work presents tight lower and upper bounds for all three types of codes for block graphs (i.e. diamond-free chordal graphs) in terms of the number of maximal cliques of a block graph and the order of the graph.

Abstract

The problems of determining the minimum-sized \emph{identifying}, \emph{locating-dominating} and \emph{open locating-dominating codes} of an input graph are special search problems that are challenging from both theoretical and computational viewpoints. In these problems, one selects a dominating set $C$ of a graph $G$ such that the vertices of a chosen subset of $V(G)$ (i.e. either $V(G)\setminus C$ or $V(G)$ itself) are uniquely determined by their neighborhoods in $C$. A typical line of attack for these problems is to determine tight bounds for the minimum codes in various graphs classes. In this work, we present tight lower and upper bounds for all three types of codes for \emph{block graphs} (i.e. diamond-free chordal graphs). Our bounds are in terms of the number of maximal cliques (or \emph{blocks}) of a block graph and the order of the graph. Two of our upper bounds verify conjectures from the literature - with one of them being now proven for block graphs in this article. As for the lower bounds, we prove them to be linear in terms of both the number of blocks and the order of the block graph. We provide examples of families of block graphs whose minimum codes attain these bounds, thus showing each bound to be tight.

Figures (7)

• Figure 1: Examples of (a) an ID-code, (b) an LD-code and (c) an OLD-code. The set of black vertices in each of the three graphs constitute the respective code of the graph.
• Figure 2: Example of different layer numbers, articulation vertices (grey) and non-articulation vertices (white) of a connected block graph.
• Figure 3: Graph $S_5 (4,3,4,5,3)$ whose LD-number attains the upper bound in Theorem \ref{['th_LD_ub']}. The black vertices represent those included in the LD-code $C$ of $G$ as described in the proof of Theorem \ref{['th_LD_ub']}.
• Figure 5: Graph $H_3$ whose LD-number attains the upper bound in Theorem \ref{['thm_twin-free']}. The black vertices represent those included in the LD-code $C^*$ of $G$ described in the proof of Theorem \ref{['thm_twin-free']}.
• Figure 6: The Bull graph $B_5$. The set of black vertices constitute an OLD-code of $B_5$.

Theorems & Definitions (28)

• Conjecture 1.1: ABLW_ICGT
• Conjecture 1.2: foucaud2016location
• Theorem 2.1
• Theorem 2.2
• Proposition 2.3
• Theorem 2.4
• Corollary 2.5
• Proposition 2.6
• Lemma 2.7
• Lemma 2.8