Generalizations of interval and proper interval graphs for simplicial complexes
Fahimeh Khosh-Ahang Ghasr
TL;DR
The paper generalizes interval and proper interval graphs to higher-dimensional simplicial complexes by introducing $d$-interval notions and associated complexes $\Delta_d(G)$ and $\mathrm{Ind}_d(G)$. It develops equivalent characterizations and hierarchies among strong, unit, and under-closed interval variants, and extends key graph-theoretic results to the simplicial setting, including the equivalence of closed and proper interval graphs. The work yields algorithmic and algebraic consequences, such as sortability criteria for $\mathrm{Ind}_d(G)$, the construction of normal Cohen-Macaulay domains from $\mathrm{Ind}_k(G)$, and forbidden-subgraph results leading to chordality and $d$-claw/$d$-paw freeness. Together, these results deepen the connections between graph theory, simplicial complexes, and commutative algebra, offering new tools for analyzing higher-dimensional independence structures.
Abstract
We introduce and investigate generalizations of interval and proper interval graphs to simplicial complexes, including strong interval, unit interval, and under closed variants. Through equivalent combinatorial and algebraic characterizations, we uncover hierarchies among these classes and extend key results to higher dimensions, such as the equivalence of closed and proper interval graphs. These formulations enable significant applications, including finding conditions for the sortability of d-independence complexes, constructions of normal Cohen-Macaulay domains linked to d-unit interval graphs, and forbidden subgraph theorems establishing chordality and d-claw-freeness. Our work advances the connections between graph theory, simplicial complexes, and commutative algebra, offering new insights into the algebraic underpinnings of combinatorial structures.