Indexing All Rooted Subgraphs of a Rooted Graph

Abstract

Let G be a connected graph in which we designate a vertex or a block (a biconnected component) as the center of G. For each cut-vertex v, let G v be the connected subgraph induced from G by v and the vertices that will be separated from the center by removal of v, where v is designated as the root of G v. We consider the set R of all such rooted subgraphs in G, and assign an integer, called an index, to each of the subgraphs so that two rooted subgraphs in R receive the same indices if and only if they are isomorphic under the constraint that their roots correspond each other. In this paper, assuming a procedure for computing a signature of each graph in a class G of biconnected graphs, we present a framework for computing indices to all rooted subgraphs of a graph G with a center which is composed of biconnected components from G. With this framework, we can find indices to all rooted subgraphs of a outerplanar graph with a center in linear time and space.

Efficient Enumeration of Stereoisomers of Outerplanar Chemical Graphs Using Dynamic Programming

“…B has no symmetry: In our previous study [10], we introduce a coding S(B) of each rooted block B. When we compute the coding, one of the two possible orientations of H(B) are detected uniquely by the graphical structure of G B .…”

“…From our previous study, we have the following theorem. This theorem means that an index (i.e., integer number) can be efficiently computed for every vertex so that two vertices have the same index if and only if the subgraphs rooted at these vertices are isomorphic.…”