Metric general position extensions of classical graph invariants

TL;DR

This paper introduces a new two-parameter framework that refines classical graph invariants by considering higher-order geodesic constraints, leading to novel insights into graph perfection and specific formulas for paths and cycles.

Contribution

It develops a general hypergraph-based framework for $k,d$-invariants, extends the theory of perfect graphs, and classifies $k,d$-perfection for paths and cycles, revealing new behaviors for higher $k$.

Findings

Exact formulas for $k,d$-chromatic number of paths and cycles.

Paths are $k,d$-perfect for all parameters.

Cycles exhibit new periodic and finite-exception behaviors for $k extgreater 2$.

Abstract

We introduce a two-parameter framework that refines several classical graph invariants by imposing higher-order constraints along bounded-length geodesics. For integers $k,d\ge1$, a vertex set is called $k,d$-independent if every shortest path of length at most $d$ contains fewer than $k$ vertices of the set, giving rise to corresponding $k,d$-independence, chromatic, clique, and domination invariants. We develop a general framework for these parameters by associating each graph with a $k$-uniform hypergraph that encodes its geodesic structure. We then establish basic bounds and monotonicity properties, and introduce a notion of $k,d$-perfection extending the classical theory of perfect graphs. Exact formulas are obtained for the $k,d$-chromatic number of paths and cycles. In particular, all paths are $k,d$-perfect for all parameters, while cycles admit a complete classification of $k,d$-perfection that recovers the classical case when $k=2$ and exhibits new periodic and finite-exception behavior for $k\ge3$. We further investigate the interaction between $k,d$-invariants and graph powers, showing that while the $k=2$ case reduces to graph powers in a straightforward way, substantially different behavior arises for higher values of $k$, even for powers of paths.