Distance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e55" altimg="si14.svg"><mml:mi>r</mml:mi></mml:math>-domination number and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e60" altimg="si14.svg"><mml:mi>r</mml:mi></mml:math>-independence complexes of graphs

Deshpande, Priyavrat; Shukla, Samir; Singh, Anurag K.

doi:10.1016/j.ejc.2022.103508

Cited by 3 publications

(2 citation statements)

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“…In another project [7] with Samir Shukla, authors have determined the homotopy type of Ind r (G) for chordal graphs G (note that class of tress is a subclass of chordal graphs). A chordal graph is a graph in which every cycle on more than 3 vertices has a chord.…”

Section: Discussionmentioning

confidence: 99%

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Higher Independence Complexes of graphs and their homotopy types

Deshpande¹,

Singh²

2020

Preprint

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For r ≥ 1, the r-independence complex of a graph G is a simplicial complex whose faces are subset I ⊆ V (G) such that each component of the induced subgraph G[I] has at most r vertices. In this article, we determine the homotopy type of r-independence complexes of certain families of graphs including complete s-partite graphs, fully whiskered graphs, cycle graphs and perfect m-ary trees. In each case, these complexes are either homotopic to a wedge of equi-dimensional spheres or are contractible. We also give a closed form formula for their homotopy types.

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Section: Discussionmentioning

confidence: 99%

“…Here, we only announce our result, without proving it. Theorem 6.1 ( [7]). The higher independence complexes of chordal graphs are either contractible or homotopy equivalent to a wedge of spheres.…”

Section: Discussionmentioning

confidence: 99%

Higher Independence Complexes of graphs and their homotopy types

Deshpande¹,

Singh²

2020

Preprint

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The Extended Dominating Sets in Graphs

Gao

Shi

et al. 2023

Asia Pac. J. Oper. Res.

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Let [Formula: see text] be a graph and let [Formula: see text] be an integer. A vertex subset [Formula: see text] is called a [Formula: see text]-extended dominating set if every vertex [Formula: see text] of [Formula: see text] satisfies one of the following conditions: the distance between [Formula: see text] and [Formula: see text] is at most one or there are at least [Formula: see text] different vertices [Formula: see text] such that the distance between [Formula: see text] and [Formula: see text] ([Formula: see text]) is two. The [Formula: see text]-extended domination number [Formula: see text] of [Formula: see text] is the minimum size over all [Formula: see text]-extended dominating sets in [Formula: see text]. When [Formula: see text], they are called the extended dominating set and the extended domination number of [Formula: see text], respectively. In this paper, we mainly study the bounds of the extended domination numbers of graphs. First, we obtain the exact values of the extended domination numbers for paths and cycles. And then the Nordhaus–Gaddum bounds for the extended domination numbers are provided. Additionally, we give some bounds of the extended domination numbers for planar graphs with small diameters. Finally, we consider the [Formula: see text]-extended domination numbers in Random graphs.

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Chordal graphs, higher independence and vertex decomposable complexes

Abdelmalek

Deshpande

Goyal

et al. 2023

Int. J. Algebra Comput.

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Given a finite simple undirected graph [Formula: see text] there is a simplicial complex [Formula: see text], called the independence complex, whose faces correspond to the independent sets of [Formula: see text]. This is a well-studied concept because it provides a fertile ground for interactions between commutative algebra, graph theory and algebraic topology. In this paper, we consider a generalization of independence complex. Given [Formula: see text], a subset of the vertex set is called [Formula: see text]-independent if the connected components of the induced subgraph have cardinality at most [Formula: see text]. The collection of all [Formula: see text]-independent subsets of [Formula: see text] form a simplicial complex called the [Formula: see text]-independence complex and is denoted by [Formula: see text]. It is known that when [Formula: see text] is a chordal graph the complex [Formula: see text] has the homotopy type of a wedge of spheres. Hence, it is natural to ask which of these complexes are shellable or even vertex decomposable. We prove, using Woodroofe’s chordal hypergraph notion, that these complexes are always shellable when the underlying chordal graph is a tree. Using the notion of vertex splittable ideals we show that for caterpillar graphs the associated [Formula: see text]-independence complex is vertex decomposable for all values of [Formula: see text]. Further, for any [Formula: see text] we construct chordal graphs on [Formula: see text] vertices such that their [Formula: see text]-independence complexes are not sequentially Cohen–Macaulay.

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Distance r-domination number and r-independence complexes of graphs

Cited by 3 publications

References 15 publications

Higher Independence Complexes of graphs and their homotopy types

Higher Independence Complexes of graphs and their homotopy types

The Extended Dominating Sets in Graphs

Chordal graphs, higher independence and vertex decomposable complexes

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