Induced universal graphs for families of small graphs

TL;DR

This paper investigates the minimal size of graphs that contain all small graphs as induced subgraphs, providing exact and heuristic algorithms, and determining the minimum number of vertices for various sizes.

Contribution

It introduces algorithms for finding induced universal graphs and determines the minimal vertex counts for small graph families, including trees.

Findings

Minimum vertex counts for k=0 to 6

Bounds for f(7) between 16 and 18

Counts of such graphs for k=0 to 5

Abstract

We present exact and heuristic algorithms that find, for a given family of graphs, a graph that contains each member of the family as an induced subgraph. For $0 \leq k \leq 6$, we give the minimum number of vertices $f(k)$ in a graph containing all $k$-vertex graphs as induced subgraphs, and show that $16 \leq f(7) \leq 18$. For $0 \leq k \leq 5$, we also give the counts of such graphs, as generated by brute-force computer search. We give additional results for small graphs containing all trees on $k$ vertices.