2016/07/15 by Pedro Montealegre, Montealegre, Pedro, Ioan Todinca +1 · 1 citation
Computer Science · #Advanced Graph Theory Research #Complexity and Algorithms in Graphs #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #Optimization and Search Problems #cs.DS
paper · pdf · doi:10.48550/arxiv.1607.04545
arxiv created 2016/07/15 · openalex publication_date 2016/07/15 · arxiv updated 2016/07/18 · openalex created_date 2022/09/28 · openalex updated_date 2026/07/28
Fomin and Villanger (STACS 2010) proved that Maximum Independent Set, Feedback Vertex Set, and more generally the problem of finding a maximum induced subgraph of treewith at most a constant t, can be solved in polynomial time on graph classes with polynomially many minimal separators. We extend these results in two directions. Let \Gpoly be the class of graphs with at most \poly(n) minimal separators, for some polynomial \poly. We show that the odd powers of a graph G have at most as many minimal separators as G. Consequently, Distance-d Independent Set, which consists in finding maximum set of vertices at pairwise distance at least d, is polynomial on \Gpoly, for any even d. The problem is NP-hard on chordal graphs for any odd d ≥ 3. We also provide polynomial algorithms for Connected Vertex Cover and Connected Feedback Vertex Set on subclasses of \Gpoly including chordal and circular-arc graphs, and we discuss variants of independent domination problems.