Graphs with Branchwidth at Most Three

In this paper we investigate both the structure of graphs with branchwidth at most three, as well as algorithms to recognise such graphs. We show that a graph has branchwidth at most three, if and only if it has treewidth at most three and does not contain the three-dimensional binary cube graph as a minor. A set of four graphs is shown to be the obstruction set of graphs with branchwidth at most three. We give a safe and complete set of reduction rules for the graphs with branchwidth at most three. Using this set, a linear time algorithm is given that checks if a given graph has branchwidth at most three, and, if so, outputs a minimum width branch decomposition.