Dominating Sets and Domination Polynomials of Paths

Alikhani, Saeid; Peng, Y. H.

doi:10.1155/2009/542040

Cited by 59 publications

(51 citation statements)

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“…(see [22]). Using tables of domination polynomials (see [23]), we think that the number of algebraic integers which can be roots of graphs with exactly four distinct domination roots are about nine numbers, but we are not able to prove it. So complete characterization of graphs with exactly four distinct domination roots remains as an open problem.…”

Section: Theorem 2 (See [17]) a Graph Has One Domination Root If Andmentioning

confidence: 95%

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Graphs Whose Certain Polynomials Have Few Distinct Roots

Alikhani

2013

ISRN Discrete Mathematics

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Let= ( , ) be a simple graph. Graph polynomials are a well-developed area useful for analyzing properties of graphs. We consider domination polynomial, matching polynomial, and edge cover polynomial of G. Graphs which their polynomials have few roots can sometimes give surprising information about the structure of the graph. This paper is primarily a survey of graphs whose domination polynomial, matching polynomial, and edge cover polynomial have few distinct roots. In addition, some new unpublished results and questions are concluded.

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Section: Theorem 2 (See [17]) a Graph Has One Domination Root If Andmentioning

confidence: 95%

“…( , ) [12][13][14]. The path 4 on 4 vertices, for example, has one dominating set of cardinality 4, four dominating sets of cardinality 3, and four dominating sets of cardinality 2; its domination polynomial is ( 4 , ) = 4 + 4 3 + 4 2 .…”

Section: Graphs With Few Domination Rootsmentioning

confidence: 99%

Graphs Whose Certain Polynomials Have Few Distinct Roots

Alikhani

2013

ISRN Discrete Mathematics

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“…For instance, using a result from [2] asserting that D(C 1 , x) = x, D(C 2 , x) = x 2 + 2x, D(C 3 , x) = x 3 + 3x 2 + 3x, and D(C n , x) = x (D(C n−1 , x) + D(C n−2 , x) + D(C n−3 , x)) for n ≥ 4, we get the following result.…”

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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“…The square of a graph: The 2 nd power of a graph with the same set of vertices as G and an edge between two vertices if and only if there is a path of length atmost 2 between them. In the next section, we construct the families of the dominating sets of the square of cycles by recursive method.…”

Section: Definitionmentioning

confidence: 99%