{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T06:31:01Z","timestamp":1768717861387,"version":"3.49.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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 $\\gamma^{\\text{ID}}(G)$. 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 $\\gamma_t^{\\text{ID}}(G)$.\r\nExtending similar characterizations for identifying codes from the literature, we characterize those graphs $G$ of order $n$ with $\\gamma_t^{\\text{ID}}(G)=n$ (the only such connected graph is $P_3$) and $\\gamma_t^{\\text{ID}}(G)=n-1$ (such graphs either satisfy $\\gamma^{\\text{ID}}(G)=n-1$ or are built from certain such graphs by adding a set of universal vertices, to each of which a private leaf is attached).\r\nThen, 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 $\\frac{3n}{4}$. 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 $C_8$ also attains the upper bound. We also provide a generalized bound for all graphs of girth at least 5 (possibly with twins).\r\nFinally, we relate $\\gamma_t^{\\text{ID}}(G)$ to the similar parameter $\\gamma^{\\text{ID}}(G)$ as well as to the location-domination number of $G$ and its variants, providing bounds that are either tight or almost tight.<\/jats:p>","DOI":"10.37236\/11342","type":"journal-article","created":{"date-parts":[[2023,7,27]],"date-time":"2023-07-27T17:14:31Z","timestamp":1690478071000},"source":"Crossref","is-referenced-by-count":2,"title":["Bounds and Extremal Graphs for Total Dominating Identifying Codes"],"prefix":"10.37236","volume":"30","author":[{"given":"Florent","family":"Foucaud","sequence":"first","affiliation":[]},{"given":"Tuomo","family":"Lehtil\u00e4","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2023,7,28]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i3p15\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i3p15\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,7,27]],"date-time":"2023-07-27T17:14:35Z","timestamp":1690478075000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v30i3p15"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,28]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,7,14]]}},"URL":"https:\/\/doi.org\/10.37236\/11342","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,28]]},"article-number":"P3.15"}}