## 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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### References:

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