Complete $$r$$ r -partite Graphs Determined by their Domination Polynomial

Anthony, Barbara M.; Picollelli, Michael E.

doi:10.1007/s00373-014-1521-2

Cited by 5 publications

(7 citation statements)

References 8 publications

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“…This graph polynomial was introduced in the paper [3] that appeared in 2014 but numerous other papers on the polynomial appeared earlier. For some very recent developments on the polynomial see [4]. From our perspective, a key information encoded into the domination polynomial is that…”

Section: On the Order Of Dominating Graphsmentioning

confidence: 99%

On the Structure of Dominating Graphs

Alikhani

Fatehi

Klavžar

2017

Graphs and Combinatorics

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The k-dominating graph D k (G) of a graph G is defined on the vertex set consisting of dominating sets of G with cardinality at most k, two such sets being adjacent if they differ by either adding or deleting a single vertex. A graph is a dominating graph if it is isomorphic to D k (G) for some graph G and some positive integer k. Answering a question of Haas and Seyffarth for graphs without isolates, it is proved that if G is such a graph of order n ≥ 2 and with G ∼ = D k (G), then k = 2 and G = K 1,n−1 for some n ≥ 4. It is also proved that for a given r there exist only a finite number of r-regular, connected dominating graphs of connected graphs. In particular, C 6 and C 8 are the only dominating graphs in the class of cycles. Some results on the order of dominating graphs are also obtained.

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Section: On the Order Of Dominating Graphsmentioning

confidence: 99%

On the Structure of Dominating Graphs

Alikhani

Fatehi

Klavžar

2017

Graphs and Combinatorics

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“…noting J r ðn; 0Þ ¼ K r ðnÞ (Anthony and Picollelli, 2015). Also let P 0 n be obtained by joining the neighbors of the two end-vertices of P n by an edge.…”

Section: Gmentioning

confidence: 99%

A new method for proving dominating uniqueness of graphs

Lau

2017

Journal of the Association of Arab Universities for Basic and A

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Let G be a graph of order n. A subset S of VðGÞ is a dominating set of G if every vertex in VðGÞ n S is adjacent to at least one vertex of S. The domination polynomial of G is the polynomial DðG; xÞ ¼ P n i¼cðGÞ dðG; iÞx i , where dðG; iÞ is the number of dominating sets of G of size i, and cðGÞ is the size of a smallest dominating set of G, called the domination number of G. We say two graphs G and H are dominating equivalent if DðG; xÞ ¼ DðH; xÞ. A graph G is said to be dominating unique, or simply D-unique, if DðH; xÞ ¼ DðG; xÞ implies that H ffi G. The goal of this paper is to find a new approach to determine the dominating uniqueness of graphs. In this paper, we define a new graph polynomial, called star polynomial, and introduced an analogy notion of star uniqueness of graphs. As an application, if G is a graph without isolated vertices, we show that a graph G is star unique if and only if G _ K m is dominating unique for each m P 0. As a by-product, the dominating uniqueness of many families of dense graphs is also determined.

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“…It is known that cycles [2] and cubic graphs of order 10 [8] (particularly, the Petersen graph) are, while if n ≡ 0(mod 3), the paths of order n are not [2]. In [9], authors completely described the complete r-partite graphs which are D-unique. Their results in the bipartite case, settles in the affirmative a conjecture in [1].…”

Section: Limits Of Domination Roots Of Friendship Graphsmentioning

confidence: 99%

On the domination polynomials of friendship graphs

Alikhani

Brown

Jahari

2016

Filomat

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Let G be a simple graph of order n. The domination polynomial of G is the polynomialis the number of dominating sets of G of size i. Let n be any positive integer and F n be the Friendship graph with 2n + 1 vertices and 3n edges, formed by the join of K 1 with nK 2 . We study the domination polynomials of this family of graphs, and in particular examine the domination roots of the family, and find the limiting curve for the roots. We also show that for every n ≥ 2, F n is not D-unique, that is, there is another non-isomorphic graph with the same domination polynomial. Also we construct some families of graphs whose real domination roots are only −2 and 0. Finally, we conclude by discussing the domination polynomials of a related family of graphs, the n-book graphs B n , formed by joining n copies of the cycle graph C 4 with a common edge.Mathematics Subject Classification: 05C31, 05C60.

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Complete $$r$$ r -partite Graphs Determined by their Domination Polynomial

Cited by 5 publications

References 8 publications

On the Structure of Dominating Graphs

On the Structure of Dominating Graphs

A new method for proving dominating uniqueness of graphs

On the domination polynomials of friendship graphs

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