Abstract
A strict confluent (SC) graph drawing is a drawing of a graph with vertices as points in the plane, where vertex adjacencies are represented not by individual curves but rather by unique smooth paths through a planar system of junctions and arcs. If all vertices of the graph lie in the outer face of the drawing, the drawing is called a strict outerconfluent (SOC) drawing. SC and SOC graphs were first considered by Eppstein et al. in Graph Drawing 2013. Here, we establish several new relationships between the class of SC graphs and other graph classes, in particular string graphs and unit-interval graphs. Further, we extend earlier results about special bipartite graph classes to the notion of strict outerconfluency, show that SOC graphs have cop number two, and establish that tree-like (\(\varDelta \)-)SOC graphs have bounded cliquewidth.
Highlights
Confluent drawings of graphs are geometric graph representations in the Euclidean plane, in which vertices are mapped to points, but edges are not drawn as individually distinguishable geometric objects
Inspired by earlier work of Gavenciak et al [19], we examine in Section 6 the cop number of strict outerconfluent (SOC) graphs and show that it is at most two
The class of string graphs [34] contains all graphs G = (V, E) which can be represented as the intersection graphs of open curves in the plane. We show that they form a superclass of strict confluent (SC) graphs and that every SOC graph is an outer-string graph [34]
Summary
Confluent drawings of graphs are geometric graph representations in the Euclidean plane, in which vertices are mapped to points, but edges are not drawn as individually distinguishable geometric objects. Since multiple edge representations may share some arcs and junctions of the drawing, this allows dense and non-planar graphs to be drawn in a plane way (e.g., see Fig. 2 for a confluent drawing of K5). Eppstein et al [14] defined the class of strict confluent (SC) drawings, which require that every edge of the graph must be represented by a unique smooth path and that there are no self-loops. They showed that for general graphs it is NP-complete to decide whether an SC drawing exists.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
