Split graphs of Dilworth number 2

Graphs with Dilworth Number Two are Pairwise Compatibility Graphs

A graph G = (V, E) is called a pairwise compatibility graph (PCG) if there exists a tree T , a positive edge-weight function w on T , and two non-negative real numbers dmin and dmax, dmin ≤ dmax, such that V coincides with the set of leaves of T , and there is an edge (u, v) ∈ E if and only if dmin ≤ dT,w(u, v) ≤ dmax where dT,w(u, v) is the sum of the weights of the edges on the unique path fr...

On pairwise compatibility graphs having Dilworth number two

A graph G = (V, E) is called a pairwise compatibility graph (PCG) if there exists a tree T , a positive edge-weight function w on T , and two non-negative real numbers dmin and dmax, dmin ≤ dmax, such that V coincides with the set of leaves of T , and there is an edge (u, v) ∈ E if and only if dmin ≤ dT,w(u, v) ≤ dmax where dT,w(u, v) is the sum of the weights of the edges on the unique path fr...

The Dilworth Number of Auto-Chordal-Bipartite Graphs

The mirror (or bipartite complement) mir(B) of a bipartite graph B = (X,Y,E) has the same color classes X and Y as B, and two vertices x ∈ X and y ∈ Y are adjacent in mir(B) if and only if xy / ∈ E. A bipartite graph is chordal bipartite if none of its induced subgraphs is a chordless cycle with at least six vertices. In this paper, we deal with chordal bipartite graphs whose mirror is chordal ...

Re ognition Algorithms for Orders of Small Width and Graphs of Small Dilworth Number

Recognizing Perfect 2-Split Graphs

A graph is a split graph if its vertices can be partitioned into a clique and a stable set. A graph is a k-split graph if its vertices can be partitioned into k sets, each of which induces a split graph. We show that the strong perfect graph conjecture is true for 2-split graphs and we design a polynomial algorithm to recognize a perfect 2-split graph.