NOTE Spectral Characterization of Tree-Width-Two Graphs

“…In Example A.11, the difference between ⌊M⌋(G) and ξ(G) occurs for very low values of these parameters, where we were able to use the characterization in [26] of all graphs having M(G) ≤ 2. Since the forbidden minors for ⌊Z + ⌋(G) = la(G) ≤ 2, namely K 4 and T 3 , are the same as the forbidden minors for ν(G) ≤ 2 [27], it follows that if ⌊M + ⌋(G) > ν(G), then ν(G) ≥ 3, and quite possibly much larger. Notice that Example A.20, which shows a difference between ⌊Z + ⌋ and ⌊M + ⌋, has ⌊M + ⌋(H) = 10, whereas the pentasun has ⌊Z⌋(H 5 ) > ⌊M⌋(H 5 ) = 2.…”

Section: Corollary 257 For Any Graph G Ccr-tw(g) = Tw(g)mentioning

confidence: 99%

Parameters Related to Tree‐Width, Zero Forcing, and Maximum Nullity of a Graph

Barioli

Barrett

Fallat

et al. 2012

Journal of Graph Theory

111

158

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Abstract. Tree-width, and variants that restrict the allowable tree decompositions, play an important role in the study of graph algorithms and have application to computer science. The zero forcing number is used to study the maximum nullity/minimum rank of the family of symmetric matrices described by a graph. We establish relationships between these parameters, including several Colin de Verdière type parameters, and introduce numerous variations, including the minor monotone floors and ceilings of some of these parameters. This leads to new graph parameters and to new characterizations of existing graph parameters. In particular, tree-width, largeur d'arborescence, path-width, and proper path-width are each characterized in terms of a minor monotone floor of a certain zero forcing parameter defined by a color change rule.

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“…With this notation we are looking for two matrices 3 ) by Lemma 2.4 it follows that fib(x) is a face of E 6 and by (13) we have that X ∈ ext E 6 . Then (17) implies that dim U ii : i ∈ [6] = 6 and thus dim U 1 = dim U 2 = 3. This implies that U 1 ∩ U 2 = {0} and thus…”

Section: Two Special Graphs: K 33 and Kmentioning

confidence: 99%

Forbidden minor characterizations for low-rank optimal solutions to semidefinite programs over the elliptope

Nagy

Laurent

Varvitsiotis

2014

Journal of Combinatorial Theory, Series B

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We study a new geometric graph parameter egd(G), defined as the smallest integer r ≥ 1 for which any partial symmetric matrix which is completable to a correlation matrix and whose entries are specified at the positions of the edges of G, can be completed to a matrix in the convex hull of correlation matrices of rank at most r. This graph parameter is motivated by its relevance to the problem of finding low rank solutions to semidefinite programs over the elliptope, and also by its relevance to the bounded rank Grothendieck constant. Indeed, egd(G) ≤ r if and only if the rank-r Grothendieck constant of G is equal to 1. We show that the parameter egd(G) is minor monotone, we identify several classes of forbidden minors for egd(G) ≤ r and we give the full characterization for the case r = 2. We also show an upper bound for egd(G) in terms of a new tree-width-like parameter la (G), defined as the smallest r for which G is a minor of the strong product of a tree and K r . We show that, for any 2-connected graph G = K 3,3 on at least 6 nodes, egd(G) ≤ 2 if and only if la (G) ≤ 2.

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NOTE Spectral Characterization of Tree-Width-Two Graphs

Cited by 7 publications

References 3 publications

Geometric representations of graphs

Geometric representations of graphs

Parameters Related to Tree‐Width, Zero Forcing, and Maximum Nullity of a Graph

Forbidden minor characterizations for low-rank optimal solutions to semidefinite programs over the elliptope

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