The H-force sets of the graphs satisfying the condition of Ore's theorem

TL;DR

This paper investigates the properties of $H$-force sets in graphs satisfying Ore's theorem, classifying such graphs based on their $H$-force number, which can be $n$, $n-2$, or $n/2$, and explores their Hamiltonian cycle enforcement.

Contribution

It provides a classification of graphs satisfying Ore's condition based on their $H$-force number, revealing possible values and structural characteristics.

Findings

$H$-force number can be $n$, $n-2$, or $n/2$ in these graphs

Classified graphs according to their $H$-force number

Characterized the structure of graphs with specific $H$-force numbers

Abstract

Let $G$ be a Hamiltonian graph with $n$ vertices. A nonempty vertex set $X\subseteq V(G)$ is called a Hamiltonian cycle enforcing set (in short, an $H$-force set) of $G$ if every $X$-cycle of $G$ (i.e., a cycle of $G$ containing all vertices of $X$) is a Hamiltonian cycle. For the graph $G$, $h(G)$ is the smallest cardinality of an $H$-force set of $G$ and call it the $H$-force number of $G$. Ore's theorem states that the graph $G$ is Hamiltonian if $d(u)+d(v)\geq n$ for every pair of nonadjacent vertices $u,v$ of $G$. In this paper, we study the $H$-force sets of the graphs satisfying the condition of Ore's theorem, show that the $H$-force number of these graphs is possibly $n$, or $n-2$, or $\frac{n}{2}$ and give a classification of these graphs due to the $H$-force number.