Characterizing subgraphs of Hamming graphs

Klavžar, Sandi; Peterin, Iztok

doi:10.1002/jgt.20084

Cited by 18 publications

(13 citation statements)

References 22 publications

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“…Theorem 7 (Coloring criterium [7]). Let G be a graph and n ∈ N. Then G has dimension at most n if and only if G has a nice coloring with n colors.…”

Section: Preliminaries and Properties Of Hamming Graphsmentioning

confidence: 99%

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Induced saturation of graphs

Axenovich

Csikós

2019

Discrete Mathematics

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A graph G is H-saturated for a graph H, if G does not contain a copy of H but adding any new edge to G results in such a copy. An H-saturated graph on a given number of vertices always exists and the properties of such graphs, for example their highest density, have been studied intensively.A graph G is H-induced-saturated if G does not have an induced subgraph isomorphic to H, but adding an edge to G from its complement or deleting an edge from G results in an induced copy of H. It is not immediate anymore that H-induced-saturated graphs exist. In fact, Martin and Smith (2012) showed that there is no P 4 -induced-saturated graph. Behrens et al. (2016) proved that if H belongs to a few simple classes of graphs such as a class of odd cycles of length at least 5, stars of size at least 2, or matchings of size at least 2, then there is an H-induced-saturated graph.This paper addresses the existence question for H-induced-saturated graphs. It is shown that Cartesian products of cliques are H-induced-saturated graphs for H in several infinite families, including large families of trees. A complete characterization of all connected graphs H for which a Cartesian product of two cliques is an H-induced-saturated graph is given. Finally, several results on induced saturation for prime graphs and families of graphs are provided.

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“…Theorem 7 (Coloring criterium [7]). Let G be a graph and n ∈ N. Then G has dimension at most n if and only if G has a nice coloring with n colors.…”

Section: Preliminaries and Properties Of Hamming Graphsmentioning

confidence: 99%

“…We showed in the beginning of the proof of Claim 3.2, that w has degree 0 in H = H − K(e), thus {{w}} is a component of H − K(e). By the choice of i, n 3−i + l i + 1 (7) = C( F 3−i ) (10) ≤ C( F i ) (7) = n i + l 3−i + 1.…”

Section: Verification Of H ∈ Bmentioning

confidence: 99%

Induced saturation of graphs

Axenovich

Csikós

2019

Discrete Mathematics

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“…Feder [6,7] later obtained similar results for 2-isometric subgraphs, weak retracts, and Cartesian prime factorization. The opposite direction was taken in [15] where subgraphs and induced subgraphs of Cartesian products of complete graphs are characterized via quotient graphs and certain edge-labelings. Another characterization of subgraphs of Cartesian products can be found in [14] using some different methods.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we use the methods introduced in [15] and we show that a graph G can be nontrivially embedded in the Cartesian product of graphs as an induced subgraph if and only if the edges of G can be labeled with at least two labels obeying two labeling conditions:…”

Section: Introductionmentioning

confidence: 99%

Characterizing Flag Graphs and Induced Subgraphs of Cartesian Product Graphs

Peterin

2004

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The vertices of the flag graph (P ) of a graded poset P are its maximal chains. Two vertices are adjacent whenever two maximal chains differ in exactly one element. In this paper we characterize induced subgraphs of Cartesian product graphs and flag graphs of graded posets. The latter class of graphs lies between isometric and induced subgraphs of Cartesian products in the embedding structure theory. Both characterization use certain edge-labelings of graphs. Mathematics Subject Classifications (2000):

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“…Another class of Cartesian S-prime graphs are so-called diagonalized Cartesian products of S-prime graphs [7], which in turn play an important role in finding approximate strong product graphs, see [4] or to find the prime factors of so-called hypergraphs [6,5]. Several interesting characterizations of (basic) S-prime graphs due to Lamprey and Barnes [13,14], Klavžar et al [11,12] and Brešar [1] are known. However, although those characterizations are established and graphs can be recognized as prime (or factorizable) w.r.t.…”

Section: Introductionmentioning

confidence: 99%

On the complexity of recognizing S-composite and S-prime graphs

Hellmuth

2013

Discrete Applied Mathematics

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S-prime graphs are graphs that cannot be represented as nontrivial subgraphs of nontrivial Cartesian products of graphs, i.e., whenever it is a subgraph of a nontrivial Cartesian product graph it is a subgraph of one the factors. A graph is S-composite if it is not S-prime. Although linear time recognition algorithms for determining whether a graph is prime or not with respect to the Cartesian product are known, it remained unknown if a similar result holds also for the recognition of S-prime and S-composite graphs.In this contribution the computational complexity of recognizing S-composite and S-prime graphs is considered. Klavžar et al. [Discr. Math. 244: 223-230 (2002)] proved that a graph is S-composite if and only if it admits a nontrivial path-k-coloring. The problem of determining whether there exists a path-kcoloring for a given graph is shown to be NP-complete even for k = 2. This in turn is utilized to show that determining whether a graph is S-composite is NP-complete and thus, determining whether a graph is S-prime is CoNP-complete. Many other problems are shown to be NP-hard, using the latter results.

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Characterizing subgraphs of Hamming graphs

Cited by 18 publications

References 22 publications

Induced saturation of graphs

Induced saturation of graphs

Characterizing Flag Graphs and Induced Subgraphs of Cartesian Product Graphs

On the complexity of recognizing S-composite and S-prime graphs

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