Degree-Associated Reconstruction Parameters of Complete Multipartite Graphs and Their Complements

Ma, Meijie; Shi, Huangping; Spinoza, Hannah; West, Douglas B.

doi:10.11650/tjm.19.2015.4850

Cited by 8 publications

(5 citation statements)

References 8 publications

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“…Ma, Shi, Spinoza and West [15] recently showed also that for all complete multipartite graphs and their complements dern is usually 2 except for some exceptions. They also pointed out that a significant difference between vertex and edge degree-associated reconstruction number is that while trivially, a graph and its complement have the same drn [4], they need not have the same value of dern.…”

Section: D(h) (Ed(g) = Ed(h))mentioning

confidence: 95%

The degree-associated edge-reconstruction number of disconnected graphs and trees

Asciak

2016

Preprint

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An edge-card of a graph G is a subgraph formed by deleting an edge.The edge-reconstruction number of a graph G, ern(G), is the minimum number of edge-cards required to determine G up to isomorphism. A da-ecard is an edge-card which also specifies the degree of the deleted edge, that is, the number of edges adjacent to it. The degree-associated edge-reconstruction number, dern(G) is the minimum number of daecards that suffice to determine the graph G. In this paper we state some known results on the edge-reconstruction number of disconnected graphs and trees. Then we investigate how the degree-associated edgereconstruction number of disconnected graphs and trees vary from their respective edge-reconstruction number. We show how we can select two da-ecards to identify caterpillars uniquely. We also show that while dern(tP n ) = 2 for n > 3, dern(tP 3 ) = 3 where P n is the path on n vertices, and that, although dern(K 1,n ) = 1, dern(S n p+1 ) = 2 where S n p+1 is a tree obtained from the star K 1,n by subdividing each edge p times. Finally we conjecture that for any tree T , dern(T ) ≤ 2.

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Section: D(h) (Ed(g) = Ed(h))mentioning

confidence: 95%

The degree-associated edge-reconstruction number of disconnected graphs and trees

Asciak

2016

Preprint

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“…The ordered pair ðdðeÞ, G À eÞ is called a degree associated edge card or da-ecard of the graph G, where d(e) (called the degree of e) is the number of edges adjacent to e in G. The edeck (daedeck) of a graph G is the collection of all ecards (da-ecards) of G. For an edge reconstructible graph G, the edge reconstruction number of G is defined to be the size of the smallest subcollection of the edeck of G which is not contained in the edeck of any other graph H, H 6 ffi G: The edge reconstruction number is known for only few classes of graphs [13,14]. For an edge reconstructible graph G from its da-edeck, the degree associated edge reconstruction number of a graph G, denoted by dern(G), is the size of the smallest subcollection of the da-edeck of G which is not contained in the da-edeck of any other graph H, H 6 ffi G: Articles [10,15] and [16] are recent papers on this parameter.…”

Section: Introductionmentioning

confidence: 99%

Degree associated edge reconstruction number of split graphs with biregular independent set is one

Mathi

Monikandan

2020

AKCE International Journal of Graphs and Combinatorics

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A degree associated edge card of a graph G is an edge deleted subgraph of G with which the degree of the deleted edge is given. The degree associated edge reconstruction number of a graph G (or dern(G)) is the size of the smallest collection of the degree associated edge cards of G that uniquely determines G. A split graph G is a graph in which the vertices can be partitioned into an independent set and a clique. We prove that the dern of all split graphs with biregular independent set is one.

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“…Conjecture that remains open. For very few classes of graphs, these edge reconstruction parameters have been determined [4,12,13,16,17].…”

Section: Introductionmentioning

confidence: 99%

Degree Associated Edge Reconstruction Parameters of Strong Double Brooms

Anushadevi¹,

Monikandan²

2018

Preprint

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An edge deleted unlabeled subgraph of a graph G is an ecard. A da-ecard specifies the degree of the deleted edge along with the ecard. The degree associated edge reconstruction number of a graph G, dern(G), is the size of the smallest collection of da-ecards of G that uniquely determines G. The adversary degree associated edge reconstruction number of a graph G, adern(G), is the minimum number k such that every collection of k da-ecards of G uniquely determines G. A strong double broom is the graph on at least 5 vertices obtained from a union of (at least two) internally vertex disjoint paths with same ends u and v by appending leaves at u and v. In particular, B(n, n, mP k ) is the strong double broom with n leaves at both the ends u and v and with m internally vertex disjoint paths of order k joining u and v. We show that dern of strong double brooms is 1 or 2. We also determine adern(B(n, n, mP k )). It is 3 in most of the cases and 1 or 2 for all the remaining cases, except adern(B(1, 1, 2P k )) = 5 for k ≥ 4.

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Degree-Associated Reconstruction Parameters of Complete Multipartite Graphs and Their Complements

Cited by 8 publications

References 8 publications

The degree-associated edge-reconstruction number of disconnected graphs and trees

The degree-associated edge-reconstruction number of disconnected graphs and trees

Degree associated edge reconstruction number of split graphs with biregular independent set is one

Degree Associated Edge Reconstruction Parameters of Strong Double Brooms

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