Cage-amalgamation graphs, a common generalization of chordal and median graphs

Brešar, Boštjan; Horvat, Aleksandra Tepeh

doi:10.1016/j.ejc.2008.09.003

Cited by 7 publications

(9 citation statements)

References 10 publications

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“…The cube polynomial was generalized to a Hamming polynomial in two different ways in [3] and [7], respectively. For a survey on the cube polynomial, its extensions and related results see [16].…”

mentioning

confidence: 99%

Cube Polynomial of Fibonacci and Lucas Cubes

Klavžar

Mollard

2011

Acta Appl Math

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The cube polynomial of a graph is the counting polynomial for the number of induced k-dimensional hypercubes (k ≥ 0). We determine the cube polynomial of Fibonacci cubes and Lucas cubes, as well as the generating functions for the sequences of these cubes. Several explicit formulas for the coefficients of these polynomials are obtained, in particular they can be expressed with convolved Fibonacci numbers. Zeros of the studied cube polynomials are explicitly determined. Consequently, the coefficients sequences of cube polynomials of Fibonacci and Lucas cubes are unimodal.

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mentioning

confidence: 99%

Cube Polynomial of Fibonacci and Lucas Cubes

Klavžar

Mollard

2011

Acta Appl Math

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“…We show that they are exactly the weakly modular graphs which do not contain K 2, 3 ,

W_{4}^{-}

and the k ‐wheels

W_{k}

for

k \geq 4

as induced subgraphs. We establish that these graphs G are exactly the cage‐amalgamation graphs as introduced by Brešar and Tepeh Horvat , that is, the graphs which can be obtained via successive gated amalgamations from Cartesian products of chordal graphs; this solves the open question raised in . This result along with definitions and preliminary observations is presented in the next section, while its proof is the contents of Section .…”

Section: Introductionmentioning

confidence: 61%

“…These graphs have been introduced by Brešar and Tepeh Horvat and are called cage‐amalgamation graphs. More precisely, the Cartesian products of connected cutvertex‐free chordal graphs were called in cages , and the graphs that can be obtained by a sequence of gated amalgamations from cages were called cage‐amalgamation graphs . It can be easily shown that cage‐amalgamation graphs are weakly modular graphs and that they do not contain induced K 2, 3 , wheels

W_{k}

, and almost‐wheels

W_{k}^{-}

(the wheel

W_{k}

is a graph obtained by connecting a single vertex—the central vertex —to all vertices of the k ‐cycle; the almost‐wheel

W_{k}^{-}

is the graph obtained from

W_{k}

by deleting a spoke (i.e., an edge between the central vertex and a vertex of the k ‐cycle), see Figure for examples.…”

Section: Preliminaries and The Characterizationsmentioning

confidence: 99%

“…Since

W_{a}

has smaller cardinality than G and since

W_{a}

is a weakly modular graph without K 2, 3 ,

W_{k}

, and

W_{k}^{-}

k \geq 4

(as a gated subgraph of G ), by induction hypothesis

W_{a}

is a cage‐amalgamation graph. Since Cartesian products and gated amalgams commute (see also Lemma 3.1 of ),

G = H □ W_{a}

is a cage‐amalgamation graph as well. Finally, suppose that for some

a \in V (H)

the set

W_{a} - U_{a}

is nonempty.…”

Section: Proof Of Theoremmentioning

confidence: 99%

See 1 more Smart Citation

Retracts of Products of Chordal Graphs

Brešar

Chalopin²,

Chepoi³

et al. 2012

J. Graph Theory

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Abstract:In this article, we characterize the graphs G that are the retracts of Cartesian products of chordal graphs. We show that they are exactly the weakly modular graphs that do not contain K 2,3 , the 4-wheel minus one spoke W − 4 , and the k-wheels W k (for k ≥ 4) as induced subgraphs. We also show that these graphs G are exactly the cage-amalgamation graphs as introduced by Brešar and Tepeh Horvat (Cage-amalgamation graphs, a common generalization of chordal and median graphs, Eur J Combin 30 (2009), 1071-1081); this solves the open question raised by these authors. Finally, we prove that replacing all products of cliques of G by products of Euclidean simplices, we obtain a polyhedral cell complex which, endowed

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“…A very natural generalization are the so-called k-chordal graphs, in which by definition the longest induced cycles are of length k. The largest common subclass of chordal and median graphs are trees, indicating the tree-like structure of both classes. On the other hand, chordal and median graphs have a common generalization through the so-called cage-amalgamation graphs [3], for which certain tree-like equalities were proven that generalize such equalities in median graphs (counting the numbers of induced hypercubes) and in chordal graphs (counting the numbers of induced cliques).…”

Section: Introductionmentioning

confidence: 99%

Almost Self-Centered Median and Chordal Graphs

Balakrishnan¹,

Brešar²,

Changat³

et al. 2012

Taiwanese J. Math.

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Abstract. Almost self-centered graphs were recently introduced as the graphs with exactly two non-central vertices. In this paper we characterize almost selfcentered graphs among median graphs and among chordal graphs. In the first case P 4 and the graphs obtained from hypercubes by attaching to them a single leaf are the only such graphs. Among chordal graph the variety of almost self-centered graph is much richer, despite the fact that their diameter is at most 3. We also discuss almost self-centered graphs among partial cubes and among k-chordal graphs, classes of graphs that generalize median and chordal graphs, respectively. Characterizations of almost self-centered graphs among these two classes seem elusive.

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Cage-amalgamation graphs, a common generalization of chordal and median graphs

Cited by 7 publications

References 10 publications

Cube Polynomial of Fibonacci and Lucas Cubes

Cube Polynomial of Fibonacci and Lucas Cubes

Retracts of Products of Chordal Graphs

Almost Self-Centered Median and Chordal Graphs

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