On full-separating sets and related codes in graphs
Dipayan Chakraborty, Annegret K. Wagler

TL;DR
This paper introduces and studies full-separating dominating and total-dominating codes in graphs, exploring their properties, bounds, and computational complexity, including NP-hardness results.
Contribution
It defines new full-separation properties for codes, analyzes their existence, bounds, relations, and establishes NP-hardness of finding minimal codes.
Findings
Minimum FD- and FTD-code sizes differ by at most one.
Deciding equality of minimum FD- and FTD-codes is NP-hard.
Bounds and extremal cases for code sizes are provided.
Abstract
A domination-based identification problem on a graph is one where the objective is to choose a subset of the vertex set of such that has both, a domination property, that is, is either a dominating or a total-dominating set of , and a separation property, that is, any two distinct vertices of must have distinct closed or open neighborhoods in . Such a set is often referred to as a code in the literature of identification problems. In this article, we introduce a new separation property, called full-separation, as it combines aspects of the two well-studied properties of closed- and open-separation. We study it in combination with both domination and total-domination and call the resulting codes full-separating dominating codes (or FD-codes for short) and full-separating total-dominating codes (or FTD-codes for short), respectively. Incidentally,…
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Taxonomy
TopicsAdvanced Graph Theory Research · Nuclear Receptors and Signaling
