Characterizing simplex graphs

TL;DR

The paper proves that simplex graphs $S(G)$ admit four equivalent characterizations: as daisy cubes, as pc-minor-free median graphs, as graphs with peripheral $\Theta$-classes, and as graphs possessing a vertex whose degree equals the isometric dimension. It achieves this via four complementary perspectives and establishes a graph-theoretic rederivation of a recent result by Betre et al. on representing abstract simplicial complexes as clique complexes under the Weak Median Property. The work also advances the understanding of daisy cubes by identifying candidate minimal forbidden pc-minors and highlighting the open problem of a complete forbidden-pc-minor characterization. Together, these results unify several known graph families (e.g., Fibonacci and Lucas cubes) under a common framework and propose concrete directions for characterizing related classes.

Abstract

The simplex graph $S(G)$ of a graph $G$ is defined as the graph whose vertices are the cliques of $G$ (including the empty set), with two vertices being adjacent if, as cliques of $G$, they differ in exactly one vertex. Simplex graphs form a subclass of median graphs and include many well-known families of graphs, such as gear graphs, Fibonacci cubes and Lucas cubes. In this paper, we characterize simplex graphs from four different perspectives: the first focuses on a graph class associated with downwards-closed sets -- namely, the daisy cubes; the second identifies all forbidden partial cube-minors of simplex graphs; the third is from the perspective of the $Θ$ equivalent classes; and the fourth explores the relationship between the maximum degree and the isometric dimension. Furthermore, very recently, Betre et al.\ [K. H. Betre, Y. X. Zhang, C. Edmond, Pure simplicial and clique complexes with a fixed number of facets, 2024, arXiv: 2411.12945v1] proved that an abstract simplicial complex (i.e., an independence system) of a finite set can be represented to a clique complex of a graph if and only if it satisfies the Weak Median Property. As a corollary, we rederive this result by using the graph-theoretical method.

Figures (4)

• Figure 1: Graph $G$ with $V(G)=\{1,2,3,4,5\}$ and its simplex graph $S(G)$. In this example, $S(G)$ is isomorphic to a 5-dimensional Fibonacci cube.
• Figure 2: Illustration for Case 2 in the proof of (iii)$\Longrightarrow$(iv).
• Figure 3: $Q_3^{-+}$ and $Q_4^{--}$.
• Figure 4: The partial cube $H^{--}_4$.

Theorems & Definitions (22)

• Proposition 1.1
• Theorem 1.2
• Proposition 1.3
• Corollary 1.4
• Theorem 1.5
• proof
• Proposition 1.6
• proof
• Remark
• Proposition 2.1