Polarity on $H$-split graphs

In their paper titled "Polarity on $H$-split graphs," authors F. Esteban Contreras Mendoza and César Hernández Cruz explore the concept of $(s,k)$-polar partitions in graph theory. An $(s,k)$-polar partition of a graph $G$ is a partition $(A,B)$ of its vertex set $V_G$, such that the induced subgraph $G[A]$ is a complete multipartite graph with at most $s$ parts and the complement of the induced subgraph $\overline{G[B]}$ is also a complete multipartite graph with at most $k$ parts. If either $s$ or $k$ is replaced by $\infty$, there are no restrictions on the number of parts in the corresponding subgraph. The authors focus specifically on two classes of graphs that generalize split graphs - which are graphs admitting a $(1,1)$-polar partition - and provide efficient algorithms to determine whether these classes admit an $(s,k)$-polar partition for any values of $s$, including infinity. They also provide upper bounds for the order of minimal $(s,k)$-polar obstructions on these families of graphs. The paper provides detailed proofs and analysis to support their findings, making it a valuable contribution to the field of graph theory. Overall, this work sheds light on how polar partitions can be used to understand and classify certain types of graphs beyond split graphs.

Exploring $(s,k)$-Polar Partitions in Graph Theory

Graph theory is a branch of mathematics that studies the properties and relationships between objects called vertices or nodes. These objects are connected by edges, which can be thought of as lines connecting two points. In their paper titled "Polarity on $H$-split graphs," authors F. Esteban Contreras Mendoza and César Hernández Cruz explore the concept of $(s,k)$-polar partitions in graph theory.

What is an $(s,k)$-Polar Partition?

An $(s,k)$-polar partition of a graph $G$ is a partition $(A,B)$ of its vertex set $V_G$, such that the induced subgraph $G[A]$ is a complete multipartite graph with at most $s$ parts and the complement of the induced subgraph $\overline{G[B]}$ is also a complete multipartite graph with at most $k$ parts. If either $s$ or $k$ is replaced by $\infty$, there are no restrictions on the number of parts in the corresponding subgraph.

The Focus: Split Graphs and Beyond

In this research paper, Contreras Mendoza and Hernández Cruz focus specifically on two classes of graphs that generalize split graphs - which are graphs admitting a $(1,1)$-polar partition - and provide efficient algorithms to determine whether these classes admit an $(s,k)$-polar partition for any values of $s$, including infinity. They also provide upper bounds for the order of minimal $(s,k)