Which rational numbers are binding numbers?

Kane, Vinay G.; Mohanty, S. P.; Straus, E. G.

doi:10.1002/jgt.3190050407

Cited by 9 publications

(6 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is more relevant to focus our attention to the problem of determining the truncated version of the binding number (and binding set) problem, namely min{bind(G), 1}.G i v e nag r a p hG, we either conclude that bind(G) ≥ 1o r ,i f bind(G) < 1, we want to find inclusionwise maximal binding set U .I fbind(G) < 1 then every binding set U of G is an independent set of G ( [8]). In the following lemma we prove some important properties of binding sets.…”

Section: Binding Number and Binding Set Of A Graphmentioning

confidence: 99%

Connection between conjunctive capacity and structural properties of graphs

Chlebík¹,

Chlebíková²

2014

Theoretical Computer Science

View full text Add to dashboard Cite

Please cite this article in press as: M. Chlebík, J. Chlebí ková, Connection between conjunctive capacity and structural properties of graphs, Theor. Comput. Sci. (2014), http://dx.doi.org/10.1016/j.tcs. 2014.04.035 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractThe investigation of the asymptotic behaviour of various graph parameters in powers of a graph is motivated by problems in information theory and extremal combinatorics. Considering various parameters and/or various notions of graph powers we can arrive at different notions of graph capacities, of which the Shannon capacity is best known. Here we study a related notion of the so-called conjunctive capacity of a graph G, C AND (G), introduced and studied by Gargano, Körner and Vaccaro. To determine C AND (G) is a convex programming problem. We show that the optimal solution to this problem is unique and describe the structure of the solution in any (simple) graph. We prove that its reciprocal value vc C (G):is an optimal solution of the newly introduced problem of Minimum Capacitary Vertex Cover that is closely related to the LPrelaxation of the Minimum Vertex Cover Problem. We also describe its close connection with the binding number/binding set of a graph, and with the strong crown decomposition of graphs.

show abstract

Section: Binding Number and Binding Set Of A Graphmentioning

confidence: 99%

Connection between conjunctive capacity and structural properties of graphs

Chlebík¹,

Chlebíková²

2014

Theoretical Computer Science

View full text Add to dashboard Cite

show abstract

“…Given a graph G, we either conclude that bind(G) ≥ 1 or, if bind(G) < 1, we want to find inclusionwise maximal binding set U . If bind(G) < 1 then every binding set U of G is an independent set of G ( [8]). In the following lemma we prove some important properties of binding sets.…”

Section: Binding Number and Binding Set Of A Graphmentioning

confidence: 99%

“…The proof of Lemma 1 Proof. (i) Let U be a binding set of G. If bind(G) < 1 then U is an independent set, as observed in [8]. Put U I = U ∩I, U H = U ∩H and U K = U ∩K.…”

Section: Appendixmentioning

confidence: 99%

On the Conjunctive Capacity of Graphs

Chlebík

Chlebíková

2013

Lecture Notes in Computer Science

View full text Add to dashboard Cite

The investigation of the asymptotic behaviour of various graph parameters in powers of a fixed graph G = (V, E) is motivated by problems in information theory and extremal combinatorics. Considering various parameters and/or various notions of graph powers we can arrive at different notions of graph capacities, of which the Shannon capacity is best known. Here we study a related notion of the so-called conjunctive capacity of a graph G, CAND(G), introduced and studied by Gargano, Körner and Vaccaro [5], [6]. To determine CAND(G) is a convex programming problem. In this paper we show that the optimal solution to this problem is unique and describe the structure of the solution in any (simple) graph. We show that its reciprocal value vcC (G) := 1 C AND (G) is an optimal solution of the newly introduced problem of Minimum Capacitary Vertex Cover that is closely related to the LP-relaxation of the Minimum Vertex Cover Problem. We also describe its close connection with the binding number/binding set of a graph, and with the strong crown decomposition of graphs introduced in [2].

show abstract

“…For concepts not defined here see [2], [3]. We enlist few results from [2], [3], [4], [5], [6] and [7] and are as follows, Proposition 1.1. (see [7]) If G is a bipartite graph, then bind(G) ≤ 1.…”

Section: Introductionmentioning

confidence: 99%

“…Proposition 1.7. (see [5]) Let G be graph with n vertices. If bind(G) < 1, then every realizing set for bind(G) is independent and |Γ(X…”

Section: Introductionmentioning

confidence: 99%

On Graphs With Binding Number One

Walikar¹,

Mulla²

2015

Int. J. of Pure and Appl. Math.

View full text Add to dashboard Cite

I. Anderson (see [1]) asked to study class of graphs with binding number one. We characterize bipartite graphs with binding number one and prove that graphs with binding number one cannot be characterized in terms of forbidden subgraphs by embedding an arbitrary graph G in a graph H with binding number one. We obtain bounds on the size of a graph with binding number one. I. Anderson [1] has defined any graph G to be binding minimal if bind (G − x) < bind (G) .

show abstract

Which rational numbers are binding numbers?

Cited by 9 publications

References 2 publications

Connection between conjunctive capacity and structural properties of graphs

Connection between conjunctive capacity and structural properties of graphs

On the Conjunctive Capacity of Graphs

On Graphs With Binding Number One

Contact Info

Product

Resources

About