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Complexity and equivalency of multiset dimension and ID-colorings

Anni Hakanen, Ismael G. Yero

Abstract

This investigation is firstly focused into showing that two metric parameters represent the same object in graph theory. That is, we prove that the multiset resolving sets and the ID-colorings of graphs are the same thing. We also consider some computational and combinatorial problems of the multiset dimension, or equivalently, the ID-number of graphs. We prove that the decision problem concerning finding the multiset dimension of graphs is NP-complete. We consider the multiset dimension of king grids and prove that it is bounded above by 4. We also give a characterization of the strong product graphs with one factor being a complete graph, and whose multiset dimension is not infinite.

Complexity and equivalency of multiset dimension and ID-colorings

Abstract

This investigation is firstly focused into showing that two metric parameters represent the same object in graph theory. That is, we prove that the multiset resolving sets and the ID-colorings of graphs are the same thing. We also consider some computational and combinatorial problems of the multiset dimension, or equivalently, the ID-number of graphs. We prove that the decision problem concerning finding the multiset dimension of graphs is NP-complete. We consider the multiset dimension of king grids and prove that it is bounded above by 4. We also give a characterization of the strong product graphs with one factor being a complete graph, and whose multiset dimension is not infinite.
Paper Structure (6 sections, 7 theorems, 2 equations, 2 figures)

This paper contains 6 sections, 7 theorems, 2 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Two equivalent metric concepts
  3. Complexity results
  4. The king grid
  5. Strong products involving a complete graph
  6. Concluding remarks

Key Result

Theorem 2.1

Let $G$ be a graph of diameter $d$. Then $S\subset V(G)$ is an ID-coloring for $G$ if and only if $S$ is a multiset resolving set for $G$.

Figures (2)

  • Figure 2: The graph $P_6 \boxtimes P_6$ with the sets $D_2(2,2)$ and $D_2(3,3)$, respectively illustrated in black. The vertices $(2,2)$ and $(3,3)$ appear in red color.
  • Figure 3: The graphs $P_5 \boxtimes P_5$ and $P_6 \boxtimes P_6$ with the sets $S$ and $S'$, respectively illustrated in black. The four digits below each vertex are the distances in the multiset representation sorted in ascending order.

Theorems & Definitions (8)

  • Theorem 2.1
  • Corollary 2.2
  • Theorem 3.1
  • Remark 4.1
  • Proposition 4.2
  • Theorem 4.3
  • Theorem 5.1
  • Corollary 5.2