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Computing a 3-role assignment is polynomial-time solvable on complementary prisms

Diane Castonguay, Elisângela S. Dias, Fernanda N. Mesquita, Julliano R. Nascimento

TL;DR

This work resolves the $3$-role assignment problem on complementary prisms $G\overline{G}$ by providing a complete structural characterization of when such a prism admits a $3$-role assignment. It demonstrates that all non-admitting cases stem from disconnected bipartite graphs and delivers a polynomial-time algorithm to decide the existence of a $3$-role assignment. The results separate the problem into bipartite and non-bipartite regimes, with explicit constructions and a forbidden-family taxonomy yielding both theoretical insight and practical decision procedures. The findings advance understanding of role assignments in network representations and have implications for efficient graph-analytic tooling on complementary prisms.

Abstract

A $r$-role assignment of a simple graph $G$ is an assignment of $r$ distinct roles to the vertices of $G$, such that two vertices with the same role have the same set of roles assigned to related vertices. Furthermore, a specific $r$-role assignment defines a role graph, in which the vertices are the distinct $r$ roles, and there is an edge between two roles whenever there are two related vertices in the graph $G$ that correspond to these roles. We consider complementary prisms, which are graphs formed from the disjoint union of the graph with its respective complement, adding the edges of a perfect matching between their corresponding vertices. In this work, we characterize the complementary prisms that do not admit a $3$-role assignment. We highlight that all of them are complementary prisms of disconnected bipartite graphs. Moreover, using our findings, we show that the problem of deciding whether a complementary prism has a $3$-role assignment can be solved in polynomial time.

Computing a 3-role assignment is polynomial-time solvable on complementary prisms

TL;DR

This work resolves the -role assignment problem on complementary prisms by providing a complete structural characterization of when such a prism admits a -role assignment. It demonstrates that all non-admitting cases stem from disconnected bipartite graphs and delivers a polynomial-time algorithm to decide the existence of a -role assignment. The results separate the problem into bipartite and non-bipartite regimes, with explicit constructions and a forbidden-family taxonomy yielding both theoretical insight and practical decision procedures. The findings advance understanding of role assignments in network representations and have implications for efficient graph-analytic tooling on complementary prisms.

Abstract

A -role assignment of a simple graph is an assignment of distinct roles to the vertices of , such that two vertices with the same role have the same set of roles assigned to related vertices. Furthermore, a specific -role assignment defines a role graph, in which the vertices are the distinct roles, and there is an edge between two roles whenever there are two related vertices in the graph that correspond to these roles. We consider complementary prisms, which are graphs formed from the disjoint union of the graph with its respective complement, adding the edges of a perfect matching between their corresponding vertices. In this work, we characterize the complementary prisms that do not admit a -role assignment. We highlight that all of them are complementary prisms of disconnected bipartite graphs. Moreover, using our findings, we show that the problem of deciding whether a complementary prism has a -role assignment can be solved in polynomial time.
Paper Structure (7 sections, 28 theorems, 26 equations, 4 figures)

This paper contains 7 sections, 28 theorems, 26 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Complementary prisms without a 3-role assignment
  4. General results
  5. Results for complementary prisms of bipartite graphs
  6. Results for complementary prisms of non-bipartite graphs
  7. Conditions to have a 3-role assignment in complementary prisms

Key Result

Lemma 3.1

The complementary prism of $K_n$, $n\geq 2$ does not have a $3$-role assignment.

Figures (4)

  • Figure 1: Possible role graphs arising from a $3$-role assignment.
  • Figure 2: A $R_{3,2}$-role assignment from the complementary prism of $C_5$.
  • Figure 3: A $R_{3,7}$-role assignment for the complementary prism of $K_2^3$.
  • Figure 4: Example of the complementary prism of a graph with $R_{3,4}$-role assignment.

Theorems & Definitions (56)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 4.1
  • proof
  • ...and 46 more