Abstract
Recently, a general framework for enumerating every non-isomorphic element in a graph class was given. Applying this framework, some graph classes have been enumerated using supercomputers, and their catalogs are provided on the web. Such graph classes include the classes of interval graphs, permutation graphs, and proper interval graphs. Last year, the enumeration algorithm for the class of Ptolemaic graphs that consists of graphs that satisfy Ptolemy inequality for the distance was investigated. They provided a polynomial time delay algorithm, but it is far from implementation. From the viewpoint of graph classes, the class is an intersection of the class of chordal graphs and the class of distance-hereditary graphs. In this paper, using the recent framework for enumerating every non-isomorphic element in a graph class, we give enumeration algorithms for the classes of distance-hereditary graphs and Ptolemaic graphs. For distance-hereditary graphs, its delay per graph is a bit slower than a previously known theoretical enumeration algorithm, however, ours is easy for implementation. In fact, although the previously known theoretical enumeration algorithm has never been implemented, we implemented our algorithm and obtained a catalog of distance-hereditary graphs of vertex numbers up to 14. We then modified the algorithm for distance-hereditary graphs to one for Ptolemaic graphs. Its delay can be the same as one for distance-hereditary graphs, which is much efficient than one proposed last year. We succeeded to enumerate Ptolemaic graphs of vertex numbers up to 15.
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Notes
- 1.
In this paper, the original graph G has its “vertices”, while corresponding tree structure \(\mathcal{T}(G)\) has its “nodes” to distinguish them.
- 2.
The DH-tree for \(K_2\) is the only exception that it is not well-defined. It can be seen as two strong twins, or one of them is a pendant of the other. From the practical point of view, it is better to initialize the queue Q by two nodes for \(K_3\) and a path (u, v, w) of three vertices.
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This work is partially supported by JSPS KAKENHI Grant Numbers 17H06287 and 18H04091.
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Yamazaki, K., Qian, M., Uehara, R. (2021). Efficient Enumeration of Non-isomorphic Distance-Hereditary Graphs and Ptolemaic Graphs. In: Uehara, R., Hong, SH., Nandy, S.C. (eds) WALCOM: Algorithms and Computation. WALCOM 2021. Lecture Notes in Computer Science(), vol 12635. Springer, Cham. https://doi.org/10.1007/978-3-030-68211-8_23
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