On the realization of a (p,s)-digraph with prescribed degrees

Chen, Wai‐Kai

doi:10.1016/0016-0032(66)90301-2

Cited by 29 publications

(20 citation statements)

References 11 publications

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“…For a given degree sequence, Fulkerson gave a system of 2 n − 1 inequalities that are satisfied if and only if the degree sequence is digraphical. The formulation that we typically use is due to Chen [3], which reduces the number of inequalities from 2 n − 1 to n when the degree sequence is in negative lexicographic order. Our consideration of threshold digraphs gives a new proof of this result.…”

Section: Digraph Realizabilitymentioning

confidence: 99%

Threshold Digraphs

Cloteaux¹,

LaMar²,

Moseman³

et al. 2014

J. RES. NATL. INST. STAN.

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Section: Digraph Realizabilitymentioning

confidence: 99%

Threshold Digraphs

Cloteaux¹,

LaMar²,

Moseman³

et al. 2014

J. RES. NATL. INST. STAN.

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“…In particular, it amounts to verify n inequalities and whether the sum of degrees is even [7]. Several variations of the graph realization problem have been investigated, such as the bipartite realization problem [10,11] and the digraph realization problem [12,13,14,15,16].…”

Section: Introductionmentioning

confidence: 99%

“…It is worth mentioning that, contrary to the approaches used in [7,8,9,10,11,12,13,14,15,16], our solution relies on nontrivial concepts studied in combinatorics on words. Indeed, if we consider the sequences of differences of consecutive elements of leaf sequences, also called the discrete derivative of the leaf sequences, then we prove that, for caterpillar graphs, the set ∆L C = {∆L C : C is a caterpillar} of discrete derivative of leaf sequences of caterpillar graphs is precisely the set of prefix normal words.…”

Section: Introductionmentioning

confidence: 99%

Leaf realization problem, caterpillar graphs and prefix normal words

Massé¹,

Carufel²,

Goupil³

et al. 2018

Theoretical Computer Science

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Given a simple graph G with n vertices and a natural number i ≤ n, let L G (i) be the maximum number of leaves that can be realized by an induced subtree T of G with i vertices. We introduce a problem that we call the leaf realization problem, which consists in deciding whether, for a given sequence of n+1 natural numbers ( 0 , 1 , . . . , n ), there exists a simple graph G with n vertices such that i = L G (i) for i = 0, 1, . . . , n. We present basic observations on the structure of these sequences for general graphs and trees. In the particular case where G is a caterpillar graph, we exhibit a bijection between the set of the discrete derivatives of the form (∆L G (i)) 1≤i≤n−3 and the set of prefix normal words.

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“…Before doing so, however, we go into a little more detail concerning the roots of the underlying graph-theoretic problems studied here. Since early computer science and algorithmic graph theory days, studies on graph realizability of degree sequences (that is, multisets of positive integers or integer pairs) have played a prominent role, being performed both for undirected graphs [13,23] as well as digraphs [7,17,24,29]. Lately, the graph modification view gained more and more attention: given a graph, can it be changed by a minimum number of graph modifications such that the resulting graph adheres to specific constraints for its degree sequence?…”

Section: Introductionmentioning

confidence: 99%

A Parameterized Algorithmics Framework for Degree Sequence Completion Problems in Directed Graphs

et al. 2018

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There has been intensive work on the parameterized complexity of the typically NPhard task to edit undirected graphs into graphs fulfilling certain given vertex degree constraints. In this work, we lift the investigations to the case of directed graphs; herein, we focus on arc insertions. To this end, we develop a general two-stage framework which consists of efficiently solving a problem-specific number problem and transferring its solution to a solution for the graph problem by applying flow computations. In this way, we obtain fixed-parameter tractability and polynomial kernelizability results, with the central parameter being the maximum vertex in-or outdegree of the output digraph. Although there are certain similarities with the much better studied undirected case, the flow computation used in the directed case seems not to work for the undirected case while f -factor computations as used in the undirected case seem not to work for the directed case.

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On the realization of a (p,s)-digraph with prescribed degrees

Cited by 29 publications

References 11 publications

Threshold Digraphs

Threshold Digraphs

Leaf realization problem, caterpillar graphs and prefix normal words

A Parameterized Algorithmics Framework for Degree Sequence Completion Problems in Directed Graphs

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