PreprintCC BY 4.0arXiv
There are no nontrivial chordal square-complementary graphs
Abstract
We study square-complementary graphs G (satisfying G2 ≅ G̅). 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.
Related
BibTeX
@misc{aboulker2026there,
title = {{There are no nontrivial chordal square-complementary graphs}},
author = {Pierre Aboulker and Martin Milanič and Peter Muršič and Miguel A. Pizaña},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28702},
eprint = {2609.28702},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28702},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - There are no nontrivial chordal square-complementary graphs
AU - Pierre Aboulker
AU - Martin Milanič
AU - Peter Muršič
AU - Miguel A. Pizaña
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28702
UR - https://arxiv.org/abs/2609.28702
AB - We study square-complementary graphs G (satisfying G^2 ≅ G̅). 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.
ER -