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

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 $\phi :E(G)\to \{1,2,\mathrm{.}\mathrm{.}\mathrm{.},\mathrm{\mid }E(G)\mathrm{\mid }\}$ is called a labeling of the graph $G$. If $G$ is a graph, $x$ is its arbitrary vertex, and $\phi$ is its arbitrary labeling, then the set $S_G(x,\varphi)\equiv{\varphi(e)/ e\in E(G), e $\text{is incident with}$ x$\}iscalledaspectrumofthevertex$x$ofthegraph$G$atitslabeling$\varphi$\mathrm{.}Foranygraph$G$ and its arbitrary labeling $\phi$, a structure of the subgraph of $G$, induced by the subset of vertices of $G$ with an interval spectrum, is described.