On Aligned Bar 1-Visibility Graphs

A graph is called a bar 1-visibility graph if its vertices can be represented as horizontal segments, called bars, and each edge corresponds to a vertical line of sight which can traverse another bar. If all bars are aligned at one side, then the graph is an aligned bar 1-visibility graph, AB1V graph for short. We consider AB1V graphs from different angles. First, we study combinatorial properties and K5 subgraphs. Then, we establish a difference between maximal and optimal AB1V graphs, where optimal AB1V graphs have the maximum of 4n− 10 edges. We show that optimal AB1V graphs can be recognized in O(n) time and prove that an AB1V representation is determined by an ordering of the bars either from left to right or by length. Finally, we introduce a new operation, called path-addition, that admits the addition of vertex-disjoint paths to a given graph and show that AB1V graphs are closed under path-addition. This distinguishes AB1V graphs from other classes of graphs. In particular, we explore the relationship to other classes of beyond-planar graphs and show that every outer 1-planar graph is an AB1V graph, whereas AB1V graphs are incomparable, e.g., to planar, k-planar, outer fan-planar, outer fan-crossing free, fan-crossing free, bar (1, j)-visibility, and RAC graphs. Submitted: May 2016 Reviewed: November 2016 Revised: December 2016 Reviewed: December 2016 Revised: December 2016 Accepted: January 2017 Final: January 2017 Published: February 2017 Article type: Regular paper Communicated by: M. Kaykobad and R. Petreschi Supported by the Deutsche Forschungsgemeinschaft (DFG), grant Br835/18-2. An extended abstract of this paper has been presented at WALCOM 2016 [11]. E-mail addresses: [email protected] (Franz J. Brandenburg)of this paper has been presented at WALCOM 2016 [11]. E-mail addresses: [email protected] (Franz J. Brandenburg) 282 Brandenburg, Esch, Neuwirth Aligned Bar 1-Visibility Graphs