Forbidden Induced Subgraphs for Bounded $p$-Intersection Number

TL;DR

This paper characterizes the minimal forbidden induced subgraphs for graphs with bounded p-intersection number, providing bounds on their size and explicit characterizations for certain parameter values.

Contribution

It establishes size bounds for minimal forbidden subgraphs and characterizes graphs in the class for specific parameters, advancing understanding of p-intersection graph classes.

Findings

Minimal forbidden induced subgraphs have order at most 3*2^{d+1}+1.

Exponential dependence on d in the size bound is necessary.

Explicit characterizations for p=d-1 and p=d-2 without isolated or universal vertices.

Abstract

A graph $G$ has $p$-intersection number at most $d$ if it is possible to assign to every vertex $u$ of $G$, a subset $S(u)$ of some ground set $U$ with $|U|=d$ in such a way that distinct vertices $u$ and $v$ of $G$ are adjacent in $G$ if and only if $|S(u)\cap S(v)|\geq p$. We show that every minimal forbidden induced subgraph for the hereditary class ${\cal G}(d,p)$ of graphs whose $p$-intersection number is at most $d$, has order at most $3\cdot 2^{d+1}+1$, and that the exponential dependence on $d$ in this upper bound is necessary. For $p\in \{ d-1,d-2\}$, we provide more explicit results characterizing the graphs in ${\cal G}(d,p)$ without isolated/universal vertices using forbidden induced subgraphs.