Distance monotone graphs and a new characterization of hypercubes

Burosch, Gustav; Havel, Ivan M.; Laborde, Jean-Marie

doi:10.1016/0012-365x(92)90696-d

Cited by 9 publications

(9 citation statements)

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“…A graph G is said to be weakly spherical, if all its intervals are weakly spherical. Theorem 2.6 (Burosch et al [3]). Let G be a distance monotone graph.…”

Section: Theorem 25 (Aïder and Aouchiche [1]) Let G Be A Graph Withmentioning

confidence: 95%

“…Let G be a distance monotone graph. Then 3 are different vertices of G such that w 1 , w 2 , w 3 are adjacent to v, then there is a vertex u in G adjacent to w 1 and w 2 but not to w 3 . (3) If (G) = 1, G is a path, and if (G) = 2, G is a cycle of even length.…”

Section: Theorem 25 (Aïder and Aouchiche [1]) Let G Be A Graph Withmentioning

confidence: 97%

“…For example, we will write I (u, v) instead of I G (u, v). Lemma 2.1 (Burosch et al [3]). For any u, v ∈ V (G), we have that…”

Section: Preliminariesmentioning

confidence: 97%

“…Interval monotone graphs (every interval is convex) and distance monotone graphs (every interval is closed) were introduced by Mulder [9] and Burosch et al [3,4], respectively. A new characterization of hypercubes was given by Burosch et al [3]: a graph G with minimum degree at least three is a hypercube if and only if G is both distance monotone and interval monotone.…”

Section: Introductionmentioning

confidence: 99%

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A characterization of the interval distance monotone graphs

Zhang

Wang

2007

Discrete Mathematics

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A simple connected graph G is said to be interval distance monotone if the interval I (u, v) between any pair of vertices u and v in G induces a distance monotone graph. Aïder and Aouchiche [Distance monotonicity and a new characterization of hypercubes, Discrete Math. 245 (2002) 55-62] proposed the following conjecture: a graph G is interval distance monotone if and only if each of its intervals is either isomorphic to a path or to a cycle or to a hypercube. In this paper we verify the conjecture.

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“…A graph G is said to be weakly spherical, if all its intervals are weakly spherical. Theorem 2.6 (Burosch et al [3]). Let G be a distance monotone graph.…”

Section: Theorem 25 (Aïder and Aouchiche [1]) Let G Be A Graph Withmentioning

confidence: 95%

Section: Theorem 25 (Aïder and Aouchiche [1]) Let G Be A Graph Withmentioning

confidence: 97%

“…For example, we will write I (u, v) instead of I G (u, v). Lemma 2.1 (Burosch et al [3]). For any u, v ∈ V (G), we have that…”

Section: Preliminariesmentioning

confidence: 97%