Let γ(G) and ι(G) be the domination and independent domination numbers of a graph G, respectively. In this paper, we define the Price of Independence of a graph G as the ratio ι(G) γ(G) . Firstly, we bound the Price of Independence by values depending on the number of vertices. Secondly, we consider the question of computing the Price of Independence of a given graph. Unfortunately, the decisio...

The independence equivalence class of a graph $G$ is the set graphs that have same polynomial as $G$. Beaton, Brown and Cameron (arXiv:1810.05317) found classes even cycles, raised problem finding odd cycles. completely solved in this paper.

This paper shows that the proof complexity (minimum computational complexity of proving formally or asymptotically) of \P 6=NP" is super-polynomial-time with respect to a theory T , which is a consistent extension of Peano Arithmetic (PA), and PTM-!-consistent, where the PTM-!-consistency is a polynomial-time Turing machine (PTM) version of !consistency. In other words, to prove \P 6=NP" (by an...

We investigate the implication problem for independence atoms X⊥Y of disjoint attribute sets X and Y on database schemata. A relation satisfies X⊥Y if for every X-value and every Y -value that occurs in the relation there is some tuple in the relation in which the X-value occurs together with the Y -value. We establish an axiomatization by a finite set of Horn rules, and derive an algorithm for...

A $k$-independent set in a connected graph is of vertices such that any two the are at distance greater than $k$ graph. The $k$-independence number graph, denoted $\alpha_k$, size largest Recent results have made use polynomials depend on spectrum to bound number. They optimized for cases $k=1,2$. There give good (and sometimes) optimal general $k$, including case $k=3$. In this paper, we provi...

In this paper we investigate the Ehrhart Theory of independence matroid polytope uniform matroids. It is proved that these polytopes have an polynomial with positive coefficients. To do that, prove indeed all half-open-hypersimplices are positive, and tile disjointly our using them.

In this paper chain cacti are considered. First, for two specific classes of chain cacti (orto-chains and meta-chains of cycles with h vertices) the recurrence relation for independence polynomial is derived. That recurrence relation is then used in deriving explicit expressions for independence number and number of maximum independent sets for such chains. Also, the recurrence relation for tot...

We provide straightforward new nonparametric methods for testing conditional independence using local polynomial quantile regression, allowing weakly dependent data. Inspired by Hausman’s (1978) speci…cation testing ideas, our methods essentially compare two collections of estimators that converge to the same limits under correct speci…cation (conditional independence) and that diverge under th...