Finding Approximate and Constrained Motifs in Graphs

One of the emerging topics in the analysis of biological networks is the inference of motifs inside a network. In the context of metabolic network analysis, a recent approach introduced in [14], represents the network as a vertex-colored graph, while a motif M is represented as a multiset of colors. An occurrence of a motif M in a vertexcolored graph G is a connected induced subgraph of G whose vertex set is colored exactly as M. We investigate three different variants of the initial problem. The first two variants, Min-Add and Min-Substitute, deal with approximate occurrences of a motif in the graph, while the third variant, Constrained Graph Motif (or CGM for short), constrains the motif to contain a given set of vertices. We investigate the classical and parameterized complexity of the three problems. We show that MinAdd and Min-Substitute are NP-hard, even when M is a set, and the graph is a tree of degree bounded by 4 in which each color appears at most twice. Moreover, we show that Min-Substitute is in FPT when parameterized by the size of M. Finally, we consider the parameterized complexity of the CGM problem, and we give a fixed-parameter algorithm for graphs of bounded treewidth, while we show that the problem is W [2]-hard, even if the input graph has diameter 2.