Minimal Graph Embeddings via Point Deletions in Steiner triple systems
Melissa A. Huggan, Svenja Huntemann, Brett Stevens
TL;DR
The paper tackles the problem of determining the smallest Steiner triple system order $v$ in which a given graph $G$ can be embedded as the available graph after point deletions in the $Nofil$ game. It develops embedding bounds and uses Skolem sequences to construct minimal or near-minimal embeddings for the complete graphs $K_a$ and stars $K_{1,a-1}$, complemented by computational experiments for the families $ar{K}_a$, $P_a$, and $C_a$. Key contributions include explicit constructions achieving minimal or near-minimal orders across several residue classes modulo $6$, along with a detailed examination of obstructions and non-monotonicity in the minimal $v$ for empty, path, and cycle graphs; Pasch-switch methods are discussed as potential improvement tools. The results show linear growth of minimal $v$ with the graph size in the families studied and lay groundwork for future work on tighter constructions and the Nim-dimension of $Nofil$.
Abstract
The game Nofil is a two-player combinatorial game in which players take turns marking points of a design such that the set of marked points does not contain a block. Equivalently, we can think of the points as being deleted from the design and points that are on singleton sets can no longer be marked. Every game play eventually results in the design becoming a graph. Previous work has shown that every graph is reachable from some Steiner triple system (STS), although the order of the constructed STS is often far from the known lower bounds. In this paper we give embeddings of complete graphs and star graphs into a $\STS$ that is minimal or very nearly meets the bounds. We further discuss possible minimal embeddings of empty graphs, paths, and cycles.