Contact Graphs of Circular Arcs
Md. Jawaherul Alam
David Eppstein
Michael Kaufmann
Stephen G. Kobourov
Sergey Pupyrev
André Schulz
Torsten Ueckerdt
erschienen in:
Algorithms and Data Structures - 14th International Symposium, WADS 2015, Victoria, BC, Canada, August 5-7, 2015. Proceedings, pp. 1-13

We study representations of graphs by contacts of circular arcs, CCA-representations for short, where the vertices are interior-disjoint circular arcs in the plane and each edge is realized by an endpoint of one arc touching the interior of another. A graph is (2, k)-sparse if every s-vertex subgraph has at most 2s−k edges, and (2, k)-tight if in addition it has exactly 2n−k edges, where n is the number of vertices. Every graph with a CCA-representation is planar and (2, 0)-sparse, and it follows from known results that for k≥3 every (2, k)-sparse graph has a CCA-representation. Hence the question of CCA-representability is open for (2, k)-sparse graphs with 0≤k≤2. We partially answer this question by computing CCA-representations for several subclasses of planar (2, 0)-sparse graphs. Next, we study CCA-representations in which each arc has an empty convex hull. We show that every plane graph of maximum degree 4 has such a representation, but that finding such a representation for a plane (2, 0)-tight graph with maximum degree 5 is NP-complete. Finally, we describe a simple algorithm for representing plane (2, 0)-sparse graphs with wedges, where each vertex is represented with a sequence of two circular arcs (straight-line segments).

@inproceedings{AEKKPSU15, author = {Md. Jawaherul Alam and David Eppstein and Michael Kaufmann and Stephen G. Kobourov and Sergey Pupyrev and Andr{\'{e}} Schulz and Torsten Ueckerdt}, editor = {Frank Dehne and J{\"{o}}rg{-}R{\"{u}}diger Sack and Ulrike Stege}, title = {Contact Graphs of Circular Arcs}, booktitle = {Algorithms and Data Structures - 14th International Symposium, {WADS} 2015, Victoria, BC, Canada, August 5-7, 2015. Proceedings}, series = {Lecture Notes in Computer Science}, volume = {9214}, pages = {1--13}, publisher = {Springer}, year = {2015}, url = {}, doi = {10.1007/978-3-319-21840-3_1} }
Christoph Doppelbauer | 10.05.2024