Parameterized Complexity of Neighborhood Problems in Graphs with no Small Cycles
Venkatesh Raman, Saket Saurabh · 2006
Abstract. We show that several problems that are hard for various parameterized complexity classes on general graphs, become fixed parameter tractable on graphs with no small cycles. More specifically, we give fixed parameter algorithms for Dominating Set, t-Vertex Cover (where we need to cover at least t edges) and several of their variants on graphs that have no triangles or cycles of length 4. These problems are known to be W [i]-hard for some i in general graphs. We also show that the Dominating Set problem is W [2]-hard in bipartite graphs and hence on triangle free graphs. In the case of Independent Set and several of its variants, we show them fixed parameter tractable even in triangle free graphs. In contrast, we show that the Dense Subgraph problem (related to the Clique problem) is W [1]-hard on graphs with no cycles of length at most 5. Finally, we give an approximation algorithm of factor Hp+1 − 1 for the 2 Dominating Set problem in graphs with no triangles or 4-cycles, where p is the size of an optimum dominating set of the graph. This improves the previous Hn − 1 factor approximation algorithm for the problem, 2 where n is the number of vertices of the input graph. 1