A1 Refereed original research article in a scientific journal

Bounds and Extremal Graphs for Total Dominating Identifying Codes




AuthorsFoucaud Florent, Lehtilä Tuomo

PublisherELECTRONIC JOURNAL OF COMBINATORICS

Publication year2023

JournalThe Electronic Journal of Combinatorics

Journal name in sourceELECTRONIC JOURNAL OF COMBINATORICS

Journal acronymELECTRON J COMB

Article numberP3.15

Volume30

Issue3

Number of pages30

ISSN1077-8926

DOIhttps://doi.org/10.37236/11342(external)

Web address https://doi.org/10.37236/11342(external)

Self-archived copy’s web addresshttps://research.utu.fi/converis/portal/detail/Publication/180765000(external)


Abstract
An identifying code C of a graph G is a dominating set of G such that any two distinct vertices of G have distinct closed neighbourhoods within C. The smallest size of an identifying code of G is denoted & gamma;ID(G). When every vertex of G also has a neighbour in C, it is said to be a total dominating identifying code of G, and the smallest size of a total dominating identifying code of G is denoted by & gamma;ID t (G). Extending similar characterizations for identifying codes from the literature, we characterize those graphs G of order n with & gamma;tID(G) = n (the only such connected graph is P3) and & gamma;tID(G) = n - 1 (such graphs either satisfy & gamma;ID(G) = n - 1 or are built from certain such graphs by adding a set of universal vertices, to each of which a private leaf is attached).Then, using bounds from the literature, we remark that any (open and closed) twin-free tree of order n has a total dominating identifying code of size at most 3n4 . This bound is tight, and we characterize the trees reaching it. Moreover, by a new proof, we show that this upper bound actually holds for the larger class of all twin-free graphs of girth at least 5. The cycle C8 also attains the upper bound. We also provide a generalized bound for all graphs of girth at least 5 (possibly with twins).Finally, we relate & gamma;tID (G) to the similar parameter & gamma;ID(G) as well as to the location-domination number of G and its variants, providing bounds that are either tight or almost tight.

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