Tootfinder

Perfect Edge Domination in $P_6$-free Graphs and in Graphs Without Efficient Edge Dominating Sets
Luciano N. Grippo, Min Chih Lin, Camila Vera
https://arxiv.org/abs/2509.04598 h…

Perfect Edge Domination in $P_6$-free Graphs and in Graphs Without Efficient Edge Dominating Sets
An edge of a graph dominates itself along with any edge that shares an endpoint with it. An efficient edge dominating set (also called a dominating induced matching, DIM) is a subset of edges such that each edge of the graph is dominated by exactly one edge in the subset. A perfect edge dominating set is a subset of edges in which every edge outside the subset is dominated by exactly one edge within it. In this article, we establish the NP-completeness of deciding whether a graph that does not …

Low degree sum-of-squares bounds for the stability number: a copositive approach
Luis Felipe Vargas, Juan C. Vera, Peter J. C. Dickinson
https://arxiv.org/abs/2509.04949 https:/…

Low degree sum-of-squares bounds for the stability number: a copositive approach
The stability number of a graph $G$, denoted as $α(G)$, is the maximum size of an independent (stable) set in $G$. Semidefinite programming (SDP) methods, which originated from Lovász's theta number and expanded through lift-and-project hierarchies as well as sums of squares (SOS) relaxations, provide powerful tools for approximating $α(G)$. We build upon the copositive formulation of $α(G)$ and introduce a novel SDP-based hierarchy of inner approximations to the copositive cone COP$_n$, …