Testing isomorphism of chordal graphs of bounded leafage is fixed-parameter tractable
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It is known that testing isomorphism of chordal graphs is as hard as the general graph isomorphism problem. Every chordal graph can be represented as the intersection graph of some subtrees of a tree. The leafage of a chordal graph, is defined to be the minimum number of leaves in the representing tree. We construct a fixed-parameter tractable algorithm testing isomorphism of chordal graphs with bounded leafage. The key point is a fixed-parameter tractable algorithm finding the automorphism group of a colored order-3 hypergraph with bounded sizes of color classes of vertices.
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