On the Circumference of 2-Connected $\mathcal{P}_{3}$-Dominated Graphs

Abstract

Let G be a connected graph. For $x, y \in V(G)$ at distance 2, we define $J(x, y) = \{u|u \in N(x) \cap N(y), N[u] \subseteq N[x] \cup N[y]\}$, and $J^{\prime}(x, y) = \{u|u \in N (x) \cap N(y)$, if $v \in N(u) \setminus (N [x] \cup N[y])$ then $(N(u) \cup N(x) \cup N(y)) \setminus \{x,y,v\} \subseteq N(v)\}$. G is quasi-claw-free$({\mathcal{QCF}})$ if it satisfies $J(x, y) \neq \emptyset$, and G is P ₃-dominated($\mathcal{P}_{3}{\mathcal{D}}$) if it satisfies $J(x,y)\cup J^{\prime} (x,y) \neq \emptyset$, for every pair (x, y) of vertices at distance 2. Certainly ${\mathcal{P}}_3 {\mathcal{D}}$ contains ${\mathcal{QCF}}$ as a subclass. In this paper, we prove that the circumference of a 2-connected P ₃-dominated graph G on n vertices is at least min $\{3\delta+2,n\}$ or $G \in {\mathcal{F}} \cup \{K_{2,3}, K_{1,1,3}\}$, moreover if $n \leq 4\delta$ then G is hamiltonian or $G \in {\mathcal{F}}\cup\{K_{2,3}, K_{1,1,3}\}$, where ${\mathcal{F}}$ is a class of 2-connected nonhamiltonian graphs.