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Cores and shells of graphs. (English) Zbl 1274.05399

In this paper author studies some structural aspects of graphs. The \(k\)-core \(C_k(G)\) of a graph \(G\) is the maximal induced subgraph \(H\subset G\) such that \(\delta (G) \geq k\), if it exists. For \(k > 0\), the \(k\)-shell of a graph \(G\) is the subgraph of \(G\) induced by the edges contained in the \(k\)-core and not contained in the \((k + 1)\)-core. The core number of a vertex \(v\) is the largest value of \(k\) such that \(v\in C_k(G)\), and the maximum core number \(\widehat {C}(G)\) of a graph \(G\) is the maximum of the core numbers of the vertices of \(G\). A graph \(G\) is \(k\)-monocore if \(\widehat {C}(G) = \delta (G)=k\).
This paper discusses some basic results on the structure of \(k\)-cores and \(k\)-shells. In particular, an operation characterization of \(2\)-monocore graphs is proved. Some applications of cores and shells to graph coloring and domination are considered, some structural properties of \(2\)-cores are shown and some applications of monocore graphs into structure and eigenvalues of graphs are discussed.

MSC:

05C75 Structural characterization of families of graphs
05C15 Coloring of graphs and hypergraphs
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
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