A Description of the Subgraph Induced at a Labeling of a Graph by the Subset of Vertices with an Interval Spectrum

Abstract

The sets of vertices and edges of an undirected, simple, finite, connected graph $G$ are denoted by $V(G)$ and $E(G)$, respectively. An arbitrary nonempty finite subset of consecutive integers is called an interval. An injective mapping $\varphi:E(G)\rightarrow \{1,2,...,|E(G)|\}$ is called a labeling of the graph $G$. If $G$ is a graph, $x$ is its arbitrary vertex, and $\varphi$ is its arbitrary labeling, then the set $S_G(x,\varphi)\equiv\{\varphi(e)/ e\in E(G), e \textrm{is incident with} x$\} is called a spectrum of the vertex $x$ of the graph $G$ at its labeling $\varphi$. For any graph $G$ and its arbitrary labeling $\varphi$, a structure of the subgraph of $G$, induced by the subset of vertices of $G$ with an interval spectrum, is described.