Degree Sequence Index Strategy

TL;DR

The paper introduces the Degree Sequence Index Strategy (DSI), a method to bound graph invariants using degree sequence indices, leading to new bounds and generalizations for various graph parameters across different graph classes.

Contribution

The paper presents the DSI strategy, a novel approach for bounding graph invariants via degree sequences, and applies it to derive new bounds and generalizations for specific graph parameters.

Findings

New bounds on k-independence and k-domination numbers.

Generalizations of the annihilation number bounds.

Extensions of results to planar, claw-free, and K_{1,r}-free graphs.

Abstract

We introduce a procedure, called the Degree Sequence Index Strategy (DSI), by which to bound graph invariants by certain indices in the ordered degree sequence. As an illustration of the DSI strategy, we show how it can be used to give new upper and lower bounds on the $k$-independence and the $k$-domination numbers. These include, among other things, a double generalization of the annihilation number, a recently introduced upper bound on the independence number. Next, we use the DSI strategy in conjunction with planarity, to generalize some results of Caro and Roddity about independence number in planar graphs. Lastly, for claw-free and $K_{1,r}$-free graphs, we use DSI to generalize some results of Faudree, Gould, Jacobson, Lesniak and Lindquester.