Degree sequences of matrogenic graphs

Marchioro, P.; Morgana, A.; Petreschi, Rossella; Simeone, Bruno

doi:10.1016/0012-365x(84)90023-2

Cited by 21 publications

(15 citation statements)

References 6 publications

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“…(d) implies (c): Assume that d is the degree sequence of a pseudo-split matrogenic graph G. By results in [32] (see also [21] and Chapter 11 in [20]), in the canonical decomposition of any matrogenic graph, the canonical components are each isomorphic to either (1) a single vertex, (2) a net or net-complement, (3) a chordless 5-cycle, or (4) the matching mK 2 or its complement, for some m. Since G is pseudo-split and hence {2K 2 , C 4 }-free, none of the canonical components has form (4), and at most one component has form (3). By Theorem 6 and Examples 2 and 3, G (d) is then the Cartesian product of transposition graphs and at most one copy of K 6,6 − 6K 2 .…”

Section: Applicationsmentioning

confidence: 94%

On realization graphs of degree sequences

Barrus

2016

Discrete Mathematics

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Given the degree sequence d of a graph, the realization graph of d is the graph having as its vertices the labeled realizations of d, with two vertices adjacent if one realization may be obtained from the other via an edge-switching operation. We describe a connection between Cartesian products in realization graphs and the canonical decomposition of degree sequences described by R.I. Tyshkevich and others. As applications, we characterize the degree sequences whose realization graphs are triangle-free graphs or hypercubes.

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Section: Applicationsmentioning

confidence: 94%

On realization graphs of degree sequences

Barrus

2016

Discrete Mathematics

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“…It is easy to see that these characterizations are special cases of Theorem 6.2. In fact, to the characterization in [25] and to another characterization in [21] of degree sequences of matrogenic graphs we may add the following: A graph G is matrogenic (or matroidal) if and only if its degree sequence satisfies the conditions of Theorem 6.3 with both (i) and (ii) holding in (a), and with the lists (t+r, (t+1) 2s+r ) and ((t+2s+r−1) 2s+r , t+2s) (and ((t + 2) 5 ), for matroidal graphs) omitted in (b).…”

Section: Degree Sequences Of Hereditary Unigraphsmentioning

confidence: 99%

Hereditary unigraphs and Erdős–Gallai equalities

Barrus

2013

Discrete Mathematics

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We give characterizations of the structure and degree sequences of hereditary unigraphs, those graphs for which every induced subgraph is the unique realization of its degree sequence. The class of hereditary unigraphs properly contains the threshold and matrogenic graphs, and the characterizations presented here naturally generalize those known for these other classes of graphs.The degree sequence characterization of hereditary unigraphs makes use of the list of values k for which the kth Erdős-Gallai inequality holds with equality for a graphic sequence. Using the canonical decomposition of Tyshkevich, we show how this list describes structure common among all realizations of an arbitrary graphic sequence.

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“…On the contrary, it is possible to find linear recognition algorithms for all the subclasses presented in Fig. 1 [3,4,6,11,12,13,14,18]. In this paper we generalize to unigraphs the pruning algorithm designed for matrogenic graphs in [12] providing a new recognition algorithm for the whole class of unigraphs.…”

Section: Introductionmentioning

confidence: 99%

“…1 [3,4,6,11,12,13,14,18]. In this paper we generalize to unigraphs the pruning algorithm designed for matrogenic graphs in [12] providing a new recognition algorithm for the whole class of unigraphs. It is to notice that the proof of our theorem is not a straightforward generalization of the proof presented in [12], as the latter one is based on the heredity of matrogenicity while this property does not hold for unigraphs.…”

Section: Introductionmentioning

confidence: 99%

Recognition of Unigraphs through Superposition of Graphs

Borri¹,

Calamoneri²,

Petreschi³

2011

JGAA

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Unigraphs are graphs uniquely determined by their own degree sequence up to isomorphism. In this paper a structural description for unigraphs is introduced: vertex set is partitioned into three disjoint sets while edge set is divided into two different classes. This characterization allows us to design a new linear time recognition algorithm that works recursively pruning the degree sequence of the graph. The algorithm detects two particular graphs whose superposition generates the given unigraph.

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Degree sequences of matrogenic graphs

Cited by 21 publications

References 6 publications

On realization graphs of degree sequences

On realization graphs of degree sequences

Hereditary unigraphs and Erdős–Gallai equalities

Recognition of Unigraphs through Superposition of Graphs

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