On two variants of split graphs : 2-unipolar graph and k-probe-split graph

• Autores: Grippo, Luciano Norberto; Moyano, Verónica A.
• Año de publicación: 2024
• Idioma: inglés
• Tipo de recurso: artículo
• Fuente: RAIRO-Operation Research. Jul. 2024; 58(4): 3597-3606; https://www.rairo-ro.org/articles/ro/abs/2024/04/contents/contents.html
• Materia: 2-unipolar graph; K-probe-split graph; Split-width; Matemáticas; Matemática Aplicada
• Institución: Universidad Nacional de General Sarmiento

Revista con referato
Fil: Grippo, Luciano Norberto. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina.
Fil: Grippo, Luciano Norberto. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina.
A graph is called split if its vertex set can be partitioned into a stable set and a clique. In this article, we studied two variants of split graphs. A graph G is polar if its vertex set can be partitioned into two sets A and B such that G[A] is a complete multipartite graph and G[B] is a disjoint union of complete graphs. A 2-unipolar graph is a polar graph G such that G[A] is a clique and G[B] is the disjoint union of complete graphs with at most two vertices. We present a minimal forbidden induced subgraph characterization for 2-unipolar graphs. In addition, we show that they can be represented as an intersection of substars of special cacti. Let G be a graph class, the G-width of a graph G is the minimum positive integer k such that there exist k independent sets N1, … , Nk such that a set F of nonedges of G, whose endpoints belong to some Ni with i = 1, … , k, can be added so that the resulting graph G' belongs to G. We say that a graph G is k-probe-G if it has G-width at most k and when G is the class of split graphs it is denominated k-probe-split. We prove that deciding, given a graph G and a positive integer k, whether G is a h-probe-split graph for some h ? k is NP-complete. Besides, a characterization by minimal forbidden induced subgraphs for 2-probe-split cographs is presented.