There are no nontrivial chordal square-complementary graphs

2026-09-23

AI Summary

This study addresses the long-standing open problem in discrete mathematics regarding whether non-trivial chordal or distance-hereditary square-complementary graphs exist. By employing graph-theoretic analysis and proof by contradiction, specifically through comparing the incomparability of vertex closed neighborhoods, this work provides the first rigorous proof that non-trivial square-complementary graphs contain no simplicial vertices, thereby precluding their chordality and distance-heredity. Consequently, this research completely resolves the aforementioned conjecture by confirming the non-existence of non-trivial chordal or distance-hereditary square-complementary graphs, ultimately refining the structural theory framework for these related graph classes.

Abstract

We study square-complementary graphs $G$ (satisfying $G^2 \cong \overline{G}$). We show that in such graphs, no two vertices have comparable closed neighborhoods. This implies that nontrivial square-complementary graphs have no simplicial vertices and are not chordal, thus solving two open problems posed in Discrete Mathematics 327 (2014) 62-75. We also show that no nontrivial square-complementary graph is distance-hereditary.

Problem

• square-complementary graphs
• chordal graphs
• simplicial vertices
• distance-hereditary graphs

Innovation

• square-complementary graphs
• chordal graphs
• simplicial vertices
• distance-hereditary graphs
• closed neighborhoods