Degree Associated Edge Reconstruction Number of Graphs with Regular Pruned Graph (original) (raw)

A Note on the Adversary Degree Associated Reconstruction Number of Graphs

Journal of Discrete Mathematics, 2013

A vertex-deleted subgraph of a graph G is called a card of G. A card of G with which the degree of the deleted vertex is also given is called a degree associated card (or dacard) of G. The degree associated reconstruction number drn (G) of a graph G is the size of the smallest collection of dacards of G that uniquely determines G. The adversary degree associated reconstruction number of a graph G, adrn(G), is the minimum number k such that every collection of k dacards of G that uniquely determines G. In this paper, we show that adrn of wheels and complete bipartite graphs on at least 4 vertices is 2 or 3.

Adversary degree associated reconstruction number of graphs

Discrete Mathematics, Algorithms and Applications, 2015

A vertex-deleted subgraph of a graph G is called a card of G. A card of G with which the degree of the deleted vertex is also given is called a degree associated card or dacard of G. The adversary degree associated reconstruction number of a graph G, adrn (G), is the minimum number k such that every collection of k dacards of G uniquely determines G. We prove that adrn (G) = 1 + min {t+1, m-t} or 1 + min {t, m - t + 2} for a graph G obtained by subdividing t edges of K1, m. We also prove that if G is a nonempty disconnected graph whose components are cycles or complete graphs, then adrn (G) is 3 or 4, while, if G is a double star whose central vertices have degrees m + 1 and n + 1(m > n ≥ 2), then adrn (G) can be as large as n + 3.

Degree Associated Reconstruction Parameters of Total Graphs

Contributions Discret. Math., 2017

A card ( ecard ) of a graph G is a subgraph formed by deleting a vertex (an edge). A dacard ( da-ecard ) specifies the degree of the deleted vertex (edge) along with the card (ecard). The degree associated reconstruction number ( degree associated edge reconstruction number ) of a graph G, drn(G) ( dern(G) ), is the minimum number of dacards (da-ecards) that uniquely determines G. In this paper, we investigate these two parameters for the total graph of certain standard graphs.

On Making a Distinguished Vertex Minimum Degree by Vertex Deletion

Lecture Notes in Computer Science, 2011

For directed and undirected graphs, we study the problem to make a distinguished vertex the unique minimum-(in)degree vertex through deletion of a minimum number of vertices. The corresponding NP-hard optimization problems are motivated by applications concerning control in elections and social network analysis. Continuing previous work for the directed case, we show that the problem is W[2]-hard when parameterized by the graph's feedback arc set number, whereas it becomes fixed-parameter tractable when combining the parameters "feedback vertex set number" and "number of vertices to delete". For the so far unstudied undirected case, we show that the problem is NP-hard and W[1]-hard when parameterized by the "number of vertices to delete". On the positive side, we show fixed-parameter tractability for several parameterizations measuring tree-likeness, including a vertex-linear problem kernel with respect to the parameter "feedback edge set number". On the contrary, we show a non-existence result concerning polynomial-size problem kernels for the combined parameter "vertex cover number and number of vertices to delete", implying corresponding nonexistence results when replacing vertex cover number by treewidth or feedback vertex set number.

On the complexity of making a distinguished vertex minimum or maximum degree by vertex deletion

Journal of Discrete Algorithms, 2015

In this paper, we investigate the approximability of two node deletion problems. Given a vertex weighted graph G = (V, E) and a specified, or "distinguished" vertex p ∈ V , MDD(min) is the problem of finding a minimum weight vertex set S ⊆ V \ {p} such that p becomes the minimum degree vertex in G[V \ S]; and MDD(max) is the problem of finding a minimum weight vertex set S ⊆ V \{p} such that p becomes the maximum degree vertex in G[V \ S]. These are known NPcomplete problems and have been studied from the parameterized complexity point of view in [1]. Here, we prove that for any ǫ > 0, both the problems cannot be approximated within a factor (1 − ǫ) log n, unless NP ⊆ Dtime(n log log n). We also show that for any ǫ > 0, MDD(min) cannot be approximated within a factor (1 − ǫ) log n on bipartite graphs, unless NP ⊆ Dtime(n log log n), and that for any ǫ > 0, MDD(max) cannot be approximated within a factor (1/2 − ǫ) log n on bipartite graphs, unless NP ⊆ Dtime(n log log n). We give an O(log n) factor approximation algorithm for MDD(max) on general graphs, provided the degree of p is O(log n). We then show that if the degree of p is n−O(log n), a similar result holds for MDD(min). We prove that MDD(max) is APX-complete on 3-regular unweighted graphs and provide an approximation algorithm with ratio 1.583 when G is a 3-regular unweighted graph. In addition, we show that MDD(min) can be solved in polynomial time when G is a regular graph of constant degree.

Degree associated reconstruction number of connected digraphs with unique end vertex

Australas. J Comb., 2016

A vertex-deleted unlabeled subdigraph of a digraph D is a card of D. A dacard specifies the degree triple (a, b, c) of the deleted vertex along with the card, where a and b are respectively the indegree and outdegree of v and c is the number of symmetric pairs of arcs (each pair considered as unordered edge) incident with v. The degree (triple) associated reconstruction number, drn(D), of a digraph D is the size of the smallest collection of dacards of D that uniquely determines D. A P-digraph is a connected digraph of order p ≥ 4 with exactly two blocks; only one of them has just two vertices and the other block has a vertex of degree triple (0, 0, p− 2) other than the cutvertex. In this paper, we prove that the drn is at most 3 for all P -digraphs except one type and show that the drn of all connected digraphs D, with a unique end vertex in D and an end vertex in D, is at most max{3, k} if the drn of the exceptional type of P-digraphs is at most k for some k.

On Making a Distinguished Vertex of Minimum Degree by Vertex Deletion

Algorithmica, 2014

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

AKCE International Journal of Graphs and Combinatorics, 2020

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.

Recognizing connectedness from vertex-deleted subgraphs

Journal of Graph Theory, 2011

Let G be a graph of order n. The vertex-deleted subgraph G −v, obtained from G by deleting the vertex v and all edges incident to v, is called a card of G. Let H be another graph of order n, disjoint from G. Then the number of common cards of G and H is the maximum number of disjoint pairs (v, w), where v and w are vertices of G and H, respectively, such that G −v ∼ = H −w. We prove that if G is connected and H is disconnected, Journal of Graph Theory ᭧ 2010 Wiley Periodicals, Inc. 285 286 JOURNAL OF GRAPH THEORY then the number of common cards of G and H is at most n / 2 +1. Thus, we can recognize the connectedness of a graph from any n / 2 +2 of its cards. Moreover, we completely characterize those pairs of graphs that attain the upper bound and show that, with the exception of six pairs of graphs of order at most 7, any pair of graphs that attains the maximum is in one of four infinite families. ᭧

Reconstruction number of graphs with unique pendant vertex

Discrete Applied Mathematics, 2020

A vertex-deleted subgraph of a graph G is called a card of G. The reconstruction number of G is the minimum number of cards of G that suffices to determine G uniquely. A P-graph (Yongzhi, 1988) is a connected graph of order p with exactly two blocks, only one of them is K 2 and the other block has a vertex of degree p − 2 other than the cut vertex. It is shown that the reconstruction number of P-graphs is three in most of the cases, which strengthens the result of Bollobas (1990). Finally, we conclude that the reconstruction number of most of the separable graphs (connected graphs with a cut vertex) with pendant vertices is three if the reconstruction number of all other connected graphs without pendant vertices is three.

Degree Associated Edge Reconstruction Number of Graphs with Regular Pruned Graph (original) (raw)

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