On Intersection Graphs of Convex Polygons

Brimkov, Valentin E.; Kafer, Sean; Szczepankiewicz, Matthew; Terhaar, Joshua

doi:10.1007/978-3-319-07148-0_4

Cited by 3 publications

(2 citation statements)

References 26 publications

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“…The maximal clique problem finds numerous applications in telecommunication and social network analysis, error-correcting codes design, fault diagnosis on large multiprocessor systems, computer vision, and pattern recognition. See [4] for a detailed presentation of the above, [6] for more references and discussion on applications of the maximal clique problem on geometric graphs to imaging sciences, and [13,26,23] and the bibliography therein for applications to research in bioinformatics and computational chemistry. The problem of enumerating all maximal cliques and finding the maximum clique is discussed in Section 4.4.…”

Section: The Number Of Maximal Cliquesmentioning

confidence: 99%

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Homothetic Polygons and Beyond: Intersection Graphs, Recognition, and Maximum Clique

Brimkov,

Junosza-Szaniawski,

Kafer

et al. 2014

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We study the Clique problem in classes of intersection graphs of convex sets in the plane. The problem is known to be NP-complete in convex-set intersection graphs and straight-line-segment intersection graphs, but solvable in polynomial time in intersection graphs of homothetic triangles. We extend the latter result by showing that for every convex polygon P with sides parallel to k directions, every nvertex graph which is an intersection graph of homothetic copies of P contains at most n k inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, the so called k DIR -CONV, which are intersection graphs of convex polygons whose sides are parallel to some fixed k directions. Moreover, we provide some lower bounds on the maximum number of maximal cliques, discuss the complexity of recognizing these classes of graphs and present relationships with other classes of convex-set intersection graphs. Finally, we generalize the upper bound on the number of maximal cliques to intersection graphs of higher-dimensional convex polytopes in Euclidean space.

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