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On non-z(mod k) dominating sets

100%

Caro Y., Jacobson M.

Discussiones Mathematicae Graph Theory

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2003

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tom 23

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nr 1

189-199

EN

For a graph G, a positive integer k, k ≥ 2, and a non-negative integer with z < k and z ≠ 1, a subset D of the vertex set V(G) is said to be a non-z (mod k) dominating set if D is a dominating set and for all x ∈ V(G), |N[x]∩D| ≢ z (mod k).For the case k = 2 and z = 0, it has been shown that these sets exist for all graphs. The problem for k ≥ 3 is unknown (the existence for even values of k and z = 0 follows from the k = 2 case.) It is the purpose of this paper to show that for k ≥ 3 and with z < k and z ≠ 1, that a non-z(mod k) dominating set exist for all trees. Also, it will be shown that for k ≥ 4, z ≥ 1, 2 or 3 that any unicyclic graph contains a non-z(mod k) dominating set. We also give a few special cases of other families of graphs for which these dominating sets must exist.

2

Odd and residue domination numbers of a graph

81%

Caro Y., Klostermeyer W., Goldwasser J.

Discussiones Mathematicae Graph Theory

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2001

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tom 21

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nr 1

119-136

EN

Let G = (V,E) be a simple, undirected graph. A set of vertices D is called an odd dominating set if |N[v] ∩ D| ≡ 1 (mod 2) for every vertex v ∈ V(G). The minimum cardinality of an odd dominating set is called the odd domination number of G, denoted by γ₁(G). In this paper, several algorithmic and structural results are presented on this parameter for grids, complements of powers of cycles, and other graph classes as well as for more general forms of "residue" domination.

3

Constrained Colouring and σ-Hypergraphs

81%

Caro Y., Lauri J., Zarb C.

Discussiones Mathematicae Graph Theory

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2015

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tom 35

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nr 1

171-189

EN

A constrained colouring or, more specifically, an (α, β)-colouring of a hypergraph H, is an assignment of colours to its vertices such that no edge of H contains less than α or more than β vertices with different colours. This notion, introduced by Bujtás and Tuza, generalises both classical hypergraph colourings and more general Voloshin colourings of hypergraphs. In fact, for r-uniform hypergraphs, classical colourings correspond to (2, r)-colourings while an important instance of Voloshin colourings of r-uniform hypergraphs gives (2, r −1)-colourings. One intriguing aspect of all these colourings, not present in classical colourings, is that H can have gaps in its (α, β)-spectrum, that is, for k1 < k2 < k3, H would be (α, β)-colourable using k1 and using k3 colours, but not using k2 colours. In an earlier paper, the first two authors introduced, for being a partition of r, a very versatile type of r-uniform hypergraph which they called -hypergraphs. They showed that, by simple manipulation of the param- eters of a σ -hypergraph H, one can obtain families of hypergraphs which have (2, r − 1)-colourings exhibiting various interesting chromatic proper- ties. They also showed that, if the smallest part of is at least 2, then H will never have a gap in its (2, r − 1)-spectrum but, quite surprisingly, they found examples where gaps re-appear when α = β = 2. In this paper we extend many of the results of the first two authors to more general (α, β)-colourings, and we study the phenomenon of the disappearance and re-appearance of gaps and show that it is not just the behaviour of a particular example but we place it within the context of a more general study of constrained colourings of σ -hypergraphs.

4

The Saturation Number for the Length of Degree Monotone Paths

81%

Caro Y., Lauri J., Zarb C.

Discussiones Mathematicae Graph Theory

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2015

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tom 35

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nr 3

557-569

EN

A degree monotone path in a graph G is a path P such that the sequence of degrees of the vertices in the order in which they appear on P is monotonic. The length (number of vertices) of the longest degree monotone path in G is denoted by mp(G). This parameter, inspired by the well-known Erdős- Szekeres theorem, has been studied by the authors in two earlier papers. Here we consider a saturation problem for the parameter mp(G). We call G saturated if, for every edge e added to G, mp(G + e) > mp(G), and we define h(n, k) to be the least possible number of edges in a saturated graph G on n vertices with mp(G) < k, while mp(G+e) ≥ k for every new edge e. We obtain linear lower and upper bounds for h(n, k), we determine exactly the values of h(n, k) for k = 3 and 4, and we present constructions of saturated graphs.

5

Connected odd dominating sets in graphs

81%

Caro Y., Klostermeyer W., Yuster R.

Discussiones Mathematicae Graph Theory

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2005

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tom 25

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nr 3

225-239

EN

An odd dominating set of a simple, undirected graph G = (V,E) is a set of vertices D ⊆ V such that |N[v] ∩ D| ≡ 1 mod 2 for all vertices v ∈ V. It is known that every graph has an odd dominating set. In this paper we consider the concept of connected odd dominating sets. We prove that the problem of deciding if a graph has a connected odd dominating set is NP-complete. We also determine the existence or non-existence of such sets in several classes of graphs. Among other results, we prove there are only 15 grid graphs that have a connected odd dominating set.

Ograniczanie wyników

5 Discussiones Mathematicae Graph Theory

5 Caro Y.

2 Klostermeyer W.

2 Lauri J.

2 Zarb C.

1 Goldwasser J.

1 Jacobson M.

1 Yuster R.

2 2015

1 2005

1 2003

1 2001

On non-z(mod k) dominating sets

Odd and residue domination numbers of a graph

Constrained Colouring and σ-Hypergraphs

The Saturation Number for the Length of Degree Monotone Paths

Connected odd dominating sets in graphs