Metric general position extensions of classical graph invariants

TL;DR

This work introduces a two-parameter generalization of fundamental graph invariants via $k,d$-independence, linking higher-order vertex constraints to geodesic structure. It builds a canonical $k$-uniform geodesic hypergraph $\mathcal{H}_{k,d}(G)$ to translate $k,d$-invariants into hypergraph terms and proves a suite of bounds, monotonicities, and a notion of $k,d$-perfection. The authors obtain exact formulas for coloring paths and cycles, classify $k,d$-perfection for cycles, and analyze the behavior of $k,d$-invariants under graph powers, revealing new phenomena for $k\ge3$ that diverge from classical distance-based theory. The paper provides a unified framework that interpolates between classical graph invariants and graph powers, opening rich directions for future study on higher-dimensional and product graphs, as well as potential analogues of classical perfect-graph theory in this extended setting.

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.

Figures (5)

• Figure 1: The set $S_{4,7}\cap P_{23}$ is a largest $4,7$-independent set in $P_{23}$.
• Figure 2: A $4,7$-proper coloring of $P_{24}$.
• Figure 3: A $4,6$-proper coloring of $C_{16}$ using maximally even sets. The red set is $J^0_{16,6}$, the yellow set is $J^0_{16,6}+1$, and the blue vertices are contained in $J^0_{16,6}+2$.
• Figure 4: (A) A 2-coloring of a graph $G$ using $4,4$-independent sets, and (B) a 3-coloring of $G^2$ using $3,2$-independent sets. Colors indicate distinct color classes.
• Figure 5: A $3,3$-proper coloring of $P_{23}^\ell$. The same function is a $5,9$-proper coloring of $P_{23}$

Theorems & Definitions (75)

• Definition 1.1
• Remark 1.2
• Definition 2.1
• Proposition 2.2
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Definition 2.5
• Proposition 2.6