Counting Graph Isomorphisms among Chordal Graphs with Restricted Clique Number

Computing Automorphism Groups of Chordal Graphs Whose Simplicial Components Are of Small Size

“…The fact that the s-tree is invariant under automorphisms suggests another possibility for developing an effective isomorphism testing. Now, we have the following question: Can we obtain a similar result to ones in [8] and [11]? More precisely, can we develop a polynomial-time isomorphism testing for chordal graphs whose s-components are of relatively small size?…”

“…The following properties of s-components were observed in [3] and [4] and were used implicitly in [8] and [11]. We can easily see these properties from the definition of simplicial nodes.…”

“…In particular, it is very natural to ask for an efficient isomorphism testing for chordal graphs whose clique number is relatively small, because we may employ a dynamic programming approach based on the clique tree or a well-known tree isomorphism algorithm. To this question, Klawe et al [8] and Nagoya [11] independently gave an affirmative answer by showing O(((ω + 1)!) 2 • n 3 ) time isomorphism testing, where ω denotes the clique number and n denotes the number of nodes.…”

Reduction Techniques for Graph Isomorphism in the Context of Width Parameters

“…For various other results this means that their isomorphism testing part can be replaced by the general theorem and only the part analyzing the graph class remains. For example in [26] and [22] it is shown that for chordal graphs of bounded clique number a tree model can be computed in cubic time. This tree model is unique up to the ordering of children and gives rise to an invariant family of sets of vertices capturing a tree-decomposition.…”

“…Such results exist for the parameters color multiplicity [18] (also known for hypergraphs [1]), eigenvalue multiplicity [13], rooted distance width [34], feedback vertex set number [25], bounded permutation distance [29], tree-depth [7] and connected path distance width [27]. For chordal graphs, tractability results are known for the parameters clique number [22], [26] and the size of simplicial components [32]. Yet, for many parameters, such as maximum degree, tree width and genus, it is not known whether there exist fixedparameter tractable algorithms solving isomorphism (see [25]).…”