# On Strict (Outer-)Confluent Graphs

## TL;DR

This paper explores strict outerconfluent (SOC) graphs, establishing their relationships with other graph classes, analyzing their structural properties, and extending previous results to bipartite and tree-like SOC graphs.

## Contribution

It introduces new relationships between SOC graphs and classes like string and unit-interval graphs, and analyzes properties such as cop number and cliquewidth.

## Key findings

- SOC graphs have relationships with string and unit-interval graphs.
- SOC graphs have cop number two.
- Tree-like SOC graphs have bounded cliquewidth.

## 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 ($\Delta$-)SOC graphs have bounded cliquewidth.