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Asteroidal Sets and Dominating Targets in Graphs
Abstract
The focus of this Ph.D. thesis is on various distance and domination properties in graphs. In particular, we prove strong results about the interactions between asteroidal sets and dominating targets. Our results add to or extend a plethora of results on these properties within the literature. We define the class of strict dominating pair graphs and show structural and algorithmic properties of this class. Notably, we prove that such graphs have diameter 3, 4, or contain an asteroidal quadruple. Then, we design an algorithm to to efficiently recognize chordal hereditary dominating pair graphs. We provide new results that describe the internal structure of these graphs, and prove that asteroidal quadruples may provide diameter bounds. Then, we extend the notion of polarity to dominating targets by defining the concept of polar targets. We investigate dominating targets in cycle graphs and show that they cannot have polar targets. Then, we provide a sufficient condition for a graph to have a polar target of size 3.
Subject Area
Mathematics|Computer science|Theoretical Mathematics|Applied Mathematics
Recommended Citation
Al-saadi, Oleksiy, "Asteroidal Sets and Dominating Targets in Graphs" (2024). ETD collection for University of Nebraska-Lincoln. AAI31301855.
https://digitalcommons.unl.edu/dissertations/AAI31301855