Fully Dynamic Recognition of Proper Circular-Arc Graphs
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We present a fully dynamic algorithm for the recognition of proper circular-arc (PCA) graphs. The allowed operations on the graph involve the insertion and removal of vertices (together with its incident edges) or edges. Edge operations cost O(logn) time, where n is the number of vertices of the graph, while vertex operations cost O(logn+d) time, where d is the degree of the modified vertex. We also show incremental and decremental algorithms that work in O(1) time per inserted or removed edge. As part of our algorithm, fully dynamic connectivity and co-connectivity algorithms that work in O(logn) time per operation are obtained. Also, an O(Δ) time algorithm for determining if a PCA representation corresponds to a co-bipartite graph is provided, where Δ is the maximum among the degrees of the vertices. When the graph is co-bipartite, a co-bipartition of each of its co-components is obtained within the same amount of time. As an application, we show how to find a minimal forbidden induced subgraph of a static graph in O(n+m) time.
KeywordsDynamic recognition Proper circular-arc graphs Round graphs Co-connectivity Minimal forbidden induced subgraphs
The author is grateful to Min Chih Lin for pointing out that the co-components can be found in O(Δ) time, and to Jayme Szwarcfiter for asking whether the HSS algorithms can be generalized so as to recognize PHCA or PCA graphs. These were key observations for beginning the research that gave life to this article.
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