Vertex-oblique graphs

Schreyer, Jens; Walther, Hans‐Otto; Mel'nikov, L. S.

doi:10.1016/j.disc.2005.11.091

Cited by 3 publications

(5 citation statements)

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“…It is known [12] that G is vertex-oblique if and only if G is, as we now show. Let the degree-set of G (the set of degrees of vertices of G, including repetitions) be D. The degree-sets of v, and its neighbourhood, are {r} and N v := {x 1 , .…”

Section: Introduction and Basic Resultsmentioning

confidence: 89%

“…While no graph can have distinct degrees, Schreyer et al [10] have constructed infinite classes of vertex-oblique graphs. In fact, their examples are super vertex-oblique, with the degree-types of G being distinct even from degree types of G.…”

Section: Introduction and Basic Resultsmentioning

confidence: 99%

“…These properties are preserved if we add a vertex y adjacent to the unique vertex v of maximum degree, and a vertex z adjacent to y ∪ (X\v). This is Construction 1 (with k = 1) from their paper [12], where they prove that the resulting graph is again supervertex-oblique. Since we can apply this procedure repeatedly, always preserving these properties, we have: Corollary 6.…”

Section: Theorem 3 For Any Dually Vertex-oblique Graph G There Is Amentioning

confidence: 88%

“…For any fixed k, Schreyer et al [10] constructed super vertex-oblique graphs that were k-connected, with k-connected complements. Our examples of dually vertex-oblique graphs have vertices of degree 2, and thus connectivity at most 2.…”

Section: Other Open Problemsmentioning

confidence: 99%

“…The converse of the corollary is not true (the matchings are a counterexample), because a restricted switch may give us a graph G ′ isomorphic to G. But it can be used to show, for example, that the super vertex-oblique graphs G 6 1 , G 7 1 , G 8 2 , G 9 2 , in [10] have unique vertex-type sequence. In particular, every degree appears exactly once in G 8 2 , except for five vertices of the same degree that induce a graph with only one edge; it can be checked that applying Construction 1 of [10] with k = 1 preserves these properties, and it is shown in that paper that the result is again connected and super vertex-oblique. We thus have: 6.…”

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confidence: 99%

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Dually vertex-oblique graphs

Farrugia

2007

Discrete Mathematics

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A vertex with neighbours of degrees d 1 · · · d r has vertex type (d 1 , . . . , d r ). A graph is vertex-oblique if each vertex has a distinct vertex type (no graph can have distinct degrees). Schreyer et al. [Vertex-oblique graphs, same proceedings] have constructed infinite classes of super vertex-oblique graphs, where the degree types of G are distinct even from the degree types of G.G is vertex-oblique iff G is; but G and G cannot be isomorphic, since self-complementary graphs always have non-trivial automorphisms. However, we show by construction that there are dually vertex-oblique graphs of order n, where the vertex-type sequence of G is the same as that of G; they exist iff n ≡ 0 or 1(mod 4), n 8, and for n 12 we can require them to be split graphs.We also show that a dually vertex-oblique graph and its complement are never the unique pair of graphs that have a particular vertex-type sequence; but there are infinitely many super vertex-oblique graphs whose vertex-type sequence is unique.

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Section: Introduction and Basic Resultsmentioning

confidence: 89%

Section: Introduction and Basic Resultsmentioning

confidence: 99%

Section: Theorem 3 For Any Dually Vertex-oblique Graph G There Is Amentioning

confidence: 88%

Section: Other Open Problemsmentioning

confidence: 99%

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confidence: 99%

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