TY - GEN

T1 - Contact graphs of circular arcs

AU - Alam, Md Jawaherul

AU - Eppstein, David

AU - Kaufmann, Michael

AU - Kobourov, Stephen G.

AU - Pupyrev, Sergey

AU - Schulz, André

AU - Ueckerdt, Torsten

N1 - Publisher Copyright:
© Springer International Publishing Switzerland 2015.
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2015

Y1 - 2015

N2 - 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 CCArepresentations 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).

AB - 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 CCArepresentations 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).

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U2 - 10.1007/978-3-319-21840-3_1

DO - 10.1007/978-3-319-21840-3_1

M3 - Conference contribution

AN - SCOPUS:84951855468

SN - 9783319218397

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 1

EP - 13

BT - Algorithms and Data Structures - 14th International Symposium, WADS 2015, Proceedings

A2 - Dehne, Frank

A2 - Sack, Jorg-Rudiger

A2 - Stege, Ulrike

PB - Springer-Verlag

T2 - 14th International Symposium on Algorithms and Data Structures, WADS 2015

Y2 - 5 August 2015 through 7 August 2015

ER -