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Vizing's conjecture and the one-half argument

100%
Hartnell B., Rall D.
Discussiones Mathematicae Graph Theory
|
1995
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tom 15
|
nr 2
205-216
EN
The domination number of a graph G is the smallest order, γ(G), of a dominating set for G. A conjecture of V. G. Vizing [5] states that for every pair of graphs G and H, γ(G☐H) ≥ γ(G)γ(H), where G☐H denotes the Cartesian product of G and H. We show that if the vertex set of G can be partitioned in a certain way then the above inequality holds for every graph H. The class of graphs G which have this type of partitioning includes those whose 2-packing number is no smaller than γ(G)-1 as well as the collection of graphs considered by Barcalkin and German in [1]. A crucial part of the proof depends on the well-known fact that the domination number of any connected graph of order at least two is no more than half its order.
2
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Analogues of cliques for oriented coloring

100%
Klostermeyer W., MacGillivray G.
Discussiones Mathematicae Graph Theory
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2004
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tom 24
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nr 3
373-387
EN
We examine subgraphs of oriented graphs in the context of oriented coloring that are analogous to cliques in traditional vertex coloring. Bounds on the sizes of these subgraphs are given for planar, outerplanar, and series-parallel graphs. In particular, the main result of the paper is that a planar graph cannot contain an induced subgraph D with more than 36 vertices such that each pair of vertices in D are joined by a directed path of length at most two.
3
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Efficient algorithms for minimal disjoint path problems on chordal graphs

88%
Gopalakrishnan C., Satyan C., Pandu Rangan C.
Discussiones Mathematicae Graph Theory
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1995
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tom 15
|
nr 2
119-145
EN
Disjoint paths have applications in establishing bottleneck-free communication between processors in a network. The problem of finding minimum delay disjoint paths in a network directly reduces to the problem of finding the minimal disjoint paths in the graph which models the network. Previous results for this problem on chordal graphs were an O(|V| |E|²) algorithm for 2 edge disjoint paths and an O(|V| |E|) algorithm for 2 vertex disjoint paths. In this paper, we give an O(|V| |E|) algorithm for 2 vertex disjoint paths and an O(|V|+|E|) algorithm for 2 edge disjoint paths, which is a significant improvement over the previous result.
4
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Maxclique and Unit Disk Characterizations of Strongly Chordal Graphs

88%
Caria P. D., McKee T. A.
Discussiones Mathematicae Graph Theory
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2014
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tom 34
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nr 3
593-602
EN
Maxcliques (maximal complete subgraphs) and unit disks (closed neighborhoods of vertices) sometime play almost interchangeable roles in graph theory. For instance, interchanging them makes two existing characterizations of chordal graphs into two new characterizations. More intriguingly, these characterizations of chordal graphs can be naturally strengthened to new characterizations of strongly chordal graphs
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