Split Decomposition via Graph-Labelled Trees

Problem Definition

Years and Authors of Summarized Original Work

2012; Gioan, Paul

2014; Gioan, Paul, Tedder, Corneil

Introduced by Cunningham and Edmonds [11], the split decomposition, also known as the join (or 1-join) decomposition, ranges among the classical graph decomposition schemes. Given a graph G = (V, E), a bipartition (A, B) of the vertex set V (with $\vert A\vert \geqslant 2$ and $\vert B\vert \geqslant 2$) is a split if there are subsets A′ ⊆ A and B′ ⊆ B, called frontiers, such that there is an edge between a vertex u ∈ A and v ∈ B if and only if u ∈ A′ and v ∈ B′ (see Fig. 1). A graph is prime if it does not contain any split. Observe that an induced cycle of length at least 5 is a prime graph. A graph is degenerate if every bipartition (A, B) with | A | ⩾2 and | B | ⩾2 is a split. It can be shown that a degenerate graphs are either cliques or stars. The split decomposition consists in recursively decompose a graph into a set of disjoint graphs \(\{G_{1},\ldots...