Efficient Enumeration of Induced Subtrees in a K-Degenerate Graph

Abstract

In this paper, we address the problem of enumerating all induced subtrees in an input $k$ -degenerate graph, where an induced subtree is an acyclic and connected induced subgraph. A graph $G = (V, E)$ is a $k$-degenerate graph if for any its induced subgraph has a vertex whose degree is less than or equal to $k$, and many real-world graphs have small degeneracies, or very close to small degeneracies. Although, the studies are on subgraphs enumeration, such as trees, paths, and matchings, but the problem addresses the subgraph enumeration, such as enumeration of subgraphs that are trees. Their induced subgraph versions have not been studied well. One of few examples is for chordless paths and cycles. Our motivation is to reduce the time complexity close to $O(1)$ for each solution. This type of optimal algorithms is proposed many subgraph classes such as trees, and spanning trees. Induced subtrees are fundamental object thus it should be studied deeply and there possibly exist some efficient algorithms. Our algorithm utilizes nice properties of $k$-degeneracy to state an effective amortized analysis. As a result, the time complexity is reduced to $O(k)$ time per induced subtree. The problem is solved in constant time for each in planar graphs, as a corollary.