A new class of reconstructible graphs from some neighbourhood conditions

Abstract

In this paper, we provide a new class of reconstructible finite graphs. We show the following theorem: Let k be a positive integer number. Let Γ be a finite graph with at least 3 vertices. Suppose that Γ satisfies the following two conditions: (i) for any two distinct vertices w,w′ ∈ V(Γ), [w,w′] ∈ E(Γ) ⬄ N(w)-{w′} ≇ N(s) for any vertex s ∈ V(Γ); (ii) there exists a vertex v ∈ V(Γ) of degree k such that for any k-vertices v₁, v₂, …, v_k ∈ V(Γ)-{v}, there exists a vertex u ∈ V(Γ) such that St²(u,Γ) ⋂ {v, v₁, v₂, …, v_k} = ∅, where N(w) is the full subgraph of Γ whose vertex set is {v ∈ V(Γ)|[w,v] ∈ E(Γ)} and St²(u,Γ) = ⋂ {St(w,Γ)| w ∈ V(St(u,Γ))}. Then the graph Γ is reconstructible. We also provide some applications and examples.