Planar median graphs and cubesquare-graphs

Seemann, Carsten R., Moulton, Vincent ORCID:, Stadler, Peter F. and Hellmuth, Marc (2023) Planar median graphs and cubesquare-graphs. Discrete Applied Mathematics, 331. pp. 38-58. ISSN 0166-218X

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Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. In this paper we provide several novel characterizations of planar median graphs. More specifically, we characterize when a planar graph G is a median graph in terms of forbidden subgraphs and the structure of isometric cycles in G, and also in terms of subgraphs of G that are contained inside and outside of 4-cycles with respect to an arbitrary planar embedding of G. These results lead us to a new characterization of planar median graphs in terms of cubesquare-graphs that is, graphs that can be obtained by starting with cubes and square-graphs, and iteratively replacing 4-cycle boundaries (relative to some embedding) by cubes or square-graphs. As a corollary we also show that a graph is planar median if and only if it can be obtained from cubes and square-graphs by a sequence of “square-boundary” amalgamations. These considerations also lead to an O(n log n)-time recognition algorithm to compute a decomposition of a planar median graph with n vertices into cubes and square-graphs.

Item Type: Article
Uncontrolled Keywords: characterization,hybercube,planar median graph,qs-graph,recognition algorithm,square-graph,discrete mathematics and combinatorics,applied mathematics ,/dk/atira/pure/subjectarea/asjc/2600/2607
Faculty \ School: Faculty of Science > School of Computing Sciences
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Depositing User: LivePure Connector
Date Deposited: 27 Jan 2023 10:31
Last Modified: 06 Feb 2023 09:30
DOI: 10.1016/j.dam.2023.01.022

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