Triangles, 4-Cycles and Parameterized (In-)Tractability

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.