A Linear Delay Algorithm for Enumeration of 2-Edge/Vertex-Connected Induced Subgraphs

In this paper, we present the first linear delay algorithms to enumerate all 2-edge-connected induced subgraphs and to enumerate all 2-vertex-connected induced subgraphs for a given simple undirected graph. We treat these subgraph enumeration problems in a more general framework based on set systems. For an element set V, (V, C subset of 2(V)) is called a set system, where we call C epsilon C a component. A nonempty subset Y subset of C is a removable set of C if C \ Y is a component and Y is a minimal removable set (MRS) of C if it is a removable set and no proper nonempty subset Z not subset of Y is a removable set of C. We say that a set system has subset-disjoint (SD) property if, for every two components C, C' epsilon C with C' not subset of C, every MRS Y of C satisfies either Y not subset of C' or Y boolean AND C' = circle divide. We assume that a set system with SD property is implicitly given by an oracle that returns an MRS of a component which is given as a query. We provide an algorithm that, given a component C, enumerates all components that are subsets of C in linear time/space with respect to | V | and oracle running time/space. We then show that, given a simple undirected graph G, the pair of the vertex set V = V (G) and the family of vertex subsets that induce 2-edge-connected (or 2-vertex-connected) subgraphs of G has SD property, where an MRS in a 2-edge-connected (or 2-vertex-connected) induced subgraph corresponds to either an ear or a single vertex with degree greater than two.