Listing maximal H-free subgraphs

Abstract

Given two graphs $G$ and $H$, where $H$ is the forbidden subgraph or pattern, $G$ is called $H$-free if no vertex subset $V^{\prime }\subseteq V\left(G\right)$ induces a subgraph $G\left[V^{\prime }\right]$ isomorphic to $H$. In the edge-induced version of the notion, $G$ is called $H$-free if no edge subset $E^{\prime }\subseteq E\left(G\right)$ induces a subgraph $G\left[E^{\prime }\right]$ isomorphic to $H$. The goal is to list all the inclusion-maximal subgraphs of $G$ that are $H$-free, according to both the edge-induced and vertex-induced versions. Apart from its theoretical interest, the problem has application in data modeling, as it corresponds to data cleaning/repairing tasks, where the entire dataset is inconsistent with respect to the constraints given in $H$, and maximal consistent portions are sought. Several output-sensitive algorithms for the vertex-induced version are presented, which depend on the constraints on $H$ and on $G$. As for the edge-induced version, we show how output-sensitive algorithms are possible for specific cases, but an efficient general technique is unlikely to exist as simply certifying a solution can be co-NP-complete.