TY - GEN

T1 - Proportional contact representations of 4-connected planar graphs

AU - Alam, Md Jawaherul

AU - Kobourov, Stephen G.

N1 - Copyright:
Copyright 2013 Elsevier B.V., All rights reserved.

PY - 2013

Y1 - 2013

N2 - In a contact representation of a planar graph, vertices are represented by interior-disjoint polygons and two polygons share a non-empty common boundary when the corresponding vertices are adjacent. In the weighted version, a weight is assigned to each vertex and a contact representation is called proportional if each polygon realizes an area proportional to the vertex weight. In this paper we study proportional contact representations of 4-connected internally triangulated planar graphs. The best known lower and upper bounds on the polygonal complexity for such graphs are 4 and 8, respectively. We narrow the gap between them by proving the existence of a representation with complexity 6. We then disprove a 10-year old conjecture on the existence of a Hamiltonian canonical cycle in a 4-connected maximal planar graph, which also implies that a previously suggested method for constructing proportional contact representations of complexity 6 for these graphs will not work. Finally we prove that it is NP-hard to decide whether a 4-connected planar graph admits a proportional contact representation using only rectangles.

AB - In a contact representation of a planar graph, vertices are represented by interior-disjoint polygons and two polygons share a non-empty common boundary when the corresponding vertices are adjacent. In the weighted version, a weight is assigned to each vertex and a contact representation is called proportional if each polygon realizes an area proportional to the vertex weight. In this paper we study proportional contact representations of 4-connected internally triangulated planar graphs. The best known lower and upper bounds on the polygonal complexity for such graphs are 4 and 8, respectively. We narrow the gap between them by proving the existence of a representation with complexity 6. We then disprove a 10-year old conjecture on the existence of a Hamiltonian canonical cycle in a 4-connected maximal planar graph, which also implies that a previously suggested method for constructing proportional contact representations of complexity 6 for these graphs will not work. Finally we prove that it is NP-hard to decide whether a 4-connected planar graph admits a proportional contact representation using only rectangles.

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U2 - 10.1007/978-3-642-36763-2_19

DO - 10.1007/978-3-642-36763-2_19

M3 - Conference contribution

AN - SCOPUS:84874180409

SN - 9783642367625

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

SP - 211

EP - 223

BT - Graph Drawing - 20th International Symposium, GD 2012, Revised Selected Papers

T2 - 20th International Symposium on Graph Drawing, GD 2012

Y2 - 19 September 2012 through 21 September 2012

ER -