Partitioning Graphs into Connected Parts
نویسندگان
چکیده
The 2-Disjoint Connected Subgraphs problem asks if a given graph has two vertex-disjoint connected subgraphs containing prespecified sets of vertices. We show that this problem is NP-complete even if one of the sets has cardinality 2. The Longest Path Contractibility problem asks for the largest integer ` for which an input graph can be contracted to the path P` on ` vertices. We show that the computational complexity of the Longest Path Contractibility problem restricted to P`-free graphs jumps from being polynomially solvable to being NP-hard at ` = 6, while this jump occurs at ` = 5 for the 2Disjoint Connected Subgraphs problem. We also present an exact algorithm that solves the 2-Disjoint Connected Subgraphs problem faster than O∗(2n) for any n-vertex P`-free graph. For ` = 6, its running time is O∗(1.5790n). We modify this algorithm to solve the Longest Path Contractibility problem for P6-free graphs inO∗(1.5790n) time.
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