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Partitioning a graph into convex sets. (English) Zbl 1223.05220

Summary: Let \(G\) be a finite simple graph. Let \(S \subseteq V(G)\), its closed interval \(I[S]\) is the set of all vertices lying on shortest paths between any pair of vertices of \(S\). The set \(S\) is convex if \(I[S]=S\). In this work we define the concept of a convex partition of graphs. If there exists a partition of \(V(G)\) into \(p\) convex sets we say that \(G\) is \(p\)-convex. We prove that it is \(NP\)-complete to decide whether a graph \(G\) is \(p\)-convex for a fixed integer \(p\geq 2\). We show that every connected chordal graph is \(p\)-convex, for \(1\leq p\leq n\). We also establish conditions on \(n\) and \(k\) to decide if the \(k\)-th power of a cycle \(C_{n}\) is \(p\)-convex. Finally, we develop a linear-time algorithm to decide if a cograph is \(p\)-convex.

MSC:

05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C38 Paths and cycles
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