Csaba P. Gabor
Abstract
The main result of this paper is an 0([V] x [E]) time algorithm for deciding whether a given graph is a circle graph, that is, the intersection graph of a set of chords on a circle. The algorithm utilizes two new graph-theoretic results, regarding necessary induced subgraphs of graphs having neither articulation points nor similar pairs of vertices. Furthermore, as a substep of the algorithm, it is shown how to find in 0([V] x [E]) time a decomposition of a graph into prime graphs, thereby improving on a result of Cunningham.
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Index Terms
- Recognizing circle graphs in polynomial time
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Reviews
Christoph M. Hoffmann
A circle graph is one that can be considered the model of a set of chords in the following way. Let the vertices of the graph be the given chords. An edge connects chord i and chord j if and only if the two chords intersect. The paper develops an O( &vbm0; V &vbm0; × &vbm0; E &vbm0; ) algorithm for recognizing whether a given graph can be considered a circle graph for some set of chords. The algorithm constructs a set of chords from the graph. At any stage, a subgraph W exists for which the corresponding chords have already been placed. The algorithm maintains a set of vertices that are adjacent to a vertex in W but have not yet been placed, and the next vertex v is chosen from these. Then v is added to W and a corresponding chord is placed. The algorithm either terminates successfully when the entire graph has been so processed, or else fails to find a suitable chord placement for v, in which case the graph is not a circle graph.
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