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On open-separating dominating codes in graphs

Dipayan Chakraborty, Annegret K. Wagler

TL;DR

The fundamental properties concerning the existence, hardness and minimality of OSD-codes are studied and the polyhedra associated with OSD-codes are discussed to discuss the polyhedra associated with OSD-codes.

Abstract

Using dominating sets to separate vertices of graphs is a well-studied problem in the larger domain of identification problems. In such problems, the objective is to choose a suitable dominating set $C$ of a graph $G$ which is also separating in the sense that the neighbourhoods of any two distinct vertices of $G$ have distinct intersections with $C$. Such a dominating and separating set $C$ of a graph is often referred to as a code in the literature. Depending on the types of dominating and separating sets used, various problems arise under various names in the literature. In this paper, we introduce a new problem in the same realm of identification problems whereby the code, called open-separating dominating code, or OD-code for short, is a dominating set and uses open neighbourhoods for separating vertices. The paper studies the fundamental properties concerning the existence, hardness and minimality of OD-codes. Due to the emergence of a close and yet difficult to establish relation of the OD-code with another well-studied code in the literature called open (neighborhood)-locating dominating code (referred to as the open-separating total-dominating code and abbreviated as OTD-code in this paper), we compare the two codes on various graph families. Finally, we also provide an equivalent reformulation of the problem of finding OD-codes of a graph as a covering problem in a suitable hypergraph and discuss the polyhedra associated with OD-codes, again in relation to OTD-codes of some graph families already studied in this context.

On open-separating dominating codes in graphs

TL;DR

The fundamental properties concerning the existence, hardness and minimality of OSD-codes are studied and the polyhedra associated with OSD-codes are discussed to discuss the polyhedra associated with OSD-codes.

Abstract

Using dominating sets to separate vertices of graphs is a well-studied problem in the larger domain of identification problems. In such problems, the objective is to choose a suitable dominating set of a graph which is also separating in the sense that the neighbourhoods of any two distinct vertices of have distinct intersections with . Such a dominating and separating set of a graph is often referred to as a code in the literature. Depending on the types of dominating and separating sets used, various problems arise under various names in the literature. In this paper, we introduce a new problem in the same realm of identification problems whereby the code, called open-separating dominating code, or OD-code for short, is a dominating set and uses open neighbourhoods for separating vertices. The paper studies the fundamental properties concerning the existence, hardness and minimality of OD-codes. Due to the emergence of a close and yet difficult to establish relation of the OD-code with another well-studied code in the literature called open (neighborhood)-locating dominating code (referred to as the open-separating total-dominating code and abbreviated as OTD-code in this paper), we compare the two codes on various graph families. Finally, we also provide an equivalent reformulation of the problem of finding OD-codes of a graph as a covering problem in a suitable hypergraph and discuss the polyhedra associated with OD-codes, again in relation to OTD-codes of some graph families already studied in this context.
Paper Structure (16 sections, 38 theorems, 14 equations, 8 figures, 1 table)

This paper contains 16 sections, 38 theorems, 14 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Reformulation, existence and bounds
  3. Hypergraph representation of the OD-problem
  4. Existence of OD-codes
  5. Bounds on OD-numbers and their relations to other X-numbers.
  6. Computational complexity of the OD-problem
  7. OD-numbers of some graph families
  8. OD-Clutters
  9. Cliques and related families
  10. Two families of bipartite graphs
  11. Families of split graphs
  12. Families of thin suns
  13. Polyhedra associated with OD-codes
  14. About covering polyhedra.
  15. Polyhedra associated with OD-codes
  16. ...and 1 more sections

Key Result

Corollary 1

The OD-hypergraph $\mathcal{H}_{{\textsc{OD}}}(G) = (V,\mathcal{F}_{{\textsc{OD}}})$ of a graph $G=(V,E)$ is composed of as hyperedges in $\mathcal{F}_{{\textsc{OD}}}$ and $\gamma^{\rm{OD}}(G)=\tau(\mathcal{H}_{{\textsc{OD}}}(G))$ holds.

Figures (8)

  • Figure 1: Minimum X-codes in a graph (the black vertices belong to the code), where (a) is both an LD- and LTD-code, (b) an OD-code, (c) an OTD-, ID-, and ITD-code.
  • Figure 2: Polynomial-time construction of the graph $G^\psi$ from an SL-SAT instance $\psi = (X,\mathcal{C})$ as in Reduction \ref{['red_NP_OD']}. The black vertices in (c) represent those in a code described in Lemma \ref{['lem_NP_1']}. The gray vertex ($v^x_1$) implies that, for some fixed variable $x=x_0 \in X$, the vertex is not included in an OD-code but is included in the OTD-code described in Lemma \ref{['lem_NP_1']}.
  • Figure 3: Three $2$-clique-stars (the square indicates the universal vertex, black vertices belong to a minimum OD-code, the grey vertex needs to be added to obtain an OTD-code), the graph in (b) is a 2-fan.
  • Figure 4: Bipartite graphs (black vertices belong to a minimum OD-code, grey vertices need to be added to obtain an OTD-code), where (a) is the $2K_2$, (b) the $P_4$, (c) the $C_6$, (d) the bow.
  • Figure 5: The black vertices depict an OD-code of the respective graph.
  • ...and 3 more figures

Theorems & Definitions (76)

  • Corollary 1
  • Corollary 2
  • Remark 1
  • proof
  • Remark 2
  • proof
  • proof
  • Theorem 1: Bondy Bondy1972
  • Lemma 1
  • proof
  • ...and 66 more