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 φ : E(G) → {1, 2, . . . , |E(G)|} is called a labeling of the graph G. If G is a graph, x is its arbitrary vertex, and φ is its arbitrary labeling, then the set SG(x, φ) ≡ {φ(e)/e ∈ E(G), e is incident with x} is called a spectrum of the vertex x of the graph G at its labeling φ. For any graph G and its arbitrary labeling φ, a structure of the subgraph of G, induced by the subset of vertices of G with an interval spectrum, is described. Mathematics Subject Classification: 05C15, 05C78