Degree Associated Reconstruction Parameters of Total Graphs (original) (raw)
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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 Edge Reconstruction Number of Graphs with Regular Pruned Graph
Electronic Journal of Graph Theory and Applications, 2015
An ecard of a graph G is a subgraph formed by deleting an edge. 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 minimum number of da-ecards 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. The maximal subgraph without end vertices of a graph G which is not a tree is the pruned graph of G. It is shown that dern of complete multipartite graphs and some connected graphs with regular pruned graph is 1 or 2. We also determine dern and adern of corona product of standard graphs.
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.
arXiv (Cornell University), 2005
In this paper we discuss reconstruction problems for graphs. We develop some new ideas like isomorphic extension of isomorphic graphs, partitioning of vertex sets into sets of equivalent points, subdeck property, etc. and develop an approach to deal with reconstruction problem. We then discuss complete sets of invariants for graphs and reconstruction conjecture. We then begin with development of few equivalent formulations of reconstruction conjecture. In the last section we briefly elaborate the formulation due to Harary its exact demand and finally proceed to give a different proof of reconstruction conjecture using reconstructibility of graph from its spanning trees and reconstructibility of tree from its pendant point deleted deck of subtrees. This last proof can be used to develop a systematic procedure to reconstruct unique graph from its deck.
On the Decomposition of Total Graphs
Obtaining a graph from any given graph is a popular area of research in Graph Theory. Concept of Total Graph falls under this category. All the vertex-vertex adjacency, vertex-edge incidence and edge-edge incidence relations are considered in the formation of the Total Graph. For a finite simple connected graph G, T(G) can be decomposed into G and complete subgraphs of order equal to the degrees of each of the vertices in G. Also, T(G) can be decomposed into disjoint union of L(G) and q copies of C 3 , where q is the size of G.
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.
A reduction of the Graph Reconstruction Conjecture
Discussiones Mathematicae Graph Theory, 2014
A graph is said to be reconstructible if it is determined up to isomorphism from the collection of all its one-vertex deleted unlabeled subgraphs. Reconstruction Conjecture (RC) asserts that all graphs on at least three vertices are reconstructible. In this paper, we prove that interval-regular graphs and some new classes of graphs are reconstructible and show that RC is true if and only if all non-geodetic and non-interval-regular blocks G with diam(G) = 2 or diam(G) = diam(G) = 3 are reconstructible.
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.