Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  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,  give fixed parameter algorithms for DOMINATING SET, t-VERTEX COVER (where  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.  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,  show them fixed parameter tractable even in triangle free graphs. In contrast,  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.
['Venkatesh Raman', 'Saket Saurabh']
I am writing a research paper on a idea Triangles, 4-cycles and parameterized (In-)tractability
[{"from": "human", "value": "I am writing a research paper on a idea Triangles, 4-cycles and parameterized (In-)tractability"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  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,  give fixed parameter algorithms for DOMINATING SET, t-VERTEX COVER (where  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.  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,  show them fixed parameter tractable even in triangle free graphs. In contrast,  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."}]