Cycle Intersection Graphs and Minimum Decycling Sets of Even Graphs

Cary, M.

doi:10.48550/arxiv.1810.04252

Cited by 2 publications

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“…Examples include total domination [27,29,34], power domination [11,16], broadcast domination [9,17,18,32,39], and geodetic domination [19], among others. In addition to these highly similar problems, more nuanced relations between domination and other areas of graph theory exist, such as decycling or network dismantling problems [6,7,10,40].…”

Section: Introductionmentioning

confidence: 99%

Dominator Colorings of Digraphs

Cary¹

2019

Preprint

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This paper serves as the first extension of the topic of dominator colorings of graphs to the setting of digraphs. We establish the dominator chromatic number over all possible orientations of paths and cycles. In this endeavor we discover that there are infinitely many counterexamples of a graph and subgraph pair for which the subgraph has a larger dominator chromatic number than the larger graph into which it embeds. Finally, a new graph invariant measuring the difference between the dominator chromatic number of a graph and the chromatic number of that graph is established and studied. The paper concludes with some of the possible avenues for extending this line of research.

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Section: Introductionmentioning

confidence: 99%

Dominator Colorings of Digraphs

Cary¹

2019

Preprint

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“…The approach this paper takes is to use cycle decompositions of Eulerian digraphs admitting simple dicycle intersection graphs. Cycle intersection graphs were introduced in [3]. The cycle intersection of an even graph G is a graph CI(G) whose vertex set is the set of cycles in a particular cycle decomposition of G and whose edges represent unique intersections (vertices) of cycles in the cycle decomposition of G. Clearly this concept can be extended to cycle decompositions for Eulerian digraphs.…”

Section: Introductionmentioning

confidence: 99%

Vertices with the second neighborhood property in Eulerian digraphs

Cary¹

2019

Opuscula Math.

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The Second Neighborhood Conjecture states that every simple digraph has a vertex whose second out-neighborhood is at least as large as its first out-neighborhood, i.e. a vertex with the Second Neighborhood Property. A cycle intersection graph of an even graph is a new graph whose vertices are the cycles in a cycle decomposition of the original graph and whose edges represent vertex intersections of the cycles. By using a digraph variant of this concept, we prove that Eulerian digraphs which admit a simple cycle intersection graph have not only adhere to the Second Neighborhood Conjecture, but that local simplicity can, in some cases, also imply the existence of a Seymour vertex in the original digraph.

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Cycle Intersection Graphs and Minimum Decycling Sets of Even Graphs

Cited by 2 publications

References 0 publications

Dominator Colorings of Digraphs

Dominator Colorings of Digraphs

Vertices with the second neighborhood property in Eulerian digraphs

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