{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:06Z","timestamp":1753893846764,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>A set $D\\subseteq V$ of vertices is said to be a (connected) distance $k$-dominating set of $G$ if the distance between each vertex $u\\in V-D$ and $D$ is at most $k$ (and $D$ induces a connected graph in $G$).  The minimum cardinality of a (connected) distance $k$-dominating set in $G$ is the (connected) distance $k$-domination number of $G$, denoted by $\\gamma_k(G)$ ($\\gamma_k^c(G)$, respectively).  The set $D$ is defined to be a total $k$-dominating set of $G$ if every vertex in $V$ is within distance $k$ from some vertex of $D$ other than itself.  The minimum cardinality among all total $k$-dominating sets of $G$ is called the total $k$-domination number of $G$ and is denoted by $\\gamma_k^t(G)$.  For $x\\in X\\subseteq V$, if $N^k[x]-N^k[X-x]\\neq\\emptyset$, the vertex $x$ is said to be $k$-irredundant in $X$.  A set $X$ containing only $k$-irredundant vertices is called $k$-irredundant.  The $k$-irredundance number of $G$, denoted by $ir_k(G)$, is the minimum cardinality taken over all maximal $k$-irredundant sets of vertices of $G$.  In this paper we establish lower bounds for the distance $k$-irredundance number of graphs and trees.  More precisely, we prove that ${5k+1\\over 2}ir_k(G)\\geq \\gamma_k^c(G)+2k$ for each connected graph $G$ and $(2k+1)ir_k(T)\\geq\\gamma_k^c(T)+2k\\geq |V|+2k-kn_1(T)$ for each tree $T=(V,E)$ with $n_1(T)$ leaves.  A class of examples shows that the latter bound is sharp.  The second inequality generalizes a result of Meierling and Volkmann and Cyman, Lema\u0144ska and Raczek regarding $\\gamma_k$ and the first generalizes a result of Favaron and Kratsch regarding $ir_1$.  Furthermore, we shall show that $\\gamma_k^c(G)\\leq{3k+1\\over2}\\gamma_k^t(G)-2k$ for each connected graph $G$, thereby generalizing a result of Favaron and Kratsch regarding $k=1$.<\/jats:p>","DOI":"10.37236\/953","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:51:03Z","timestamp":1578718263000},"source":"Crossref","is-referenced-by-count":12,"title":["Distance Domination and Distance Irredundance in Graphs"],"prefix":"10.37236","volume":"14","author":[{"given":"Adriana","family":"Hansberg","sequence":"first","affiliation":[]},{"given":"Dirk","family":"Meierling","sequence":"additional","affiliation":[]},{"given":"Lutz","family":"Volkmann","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2007,5,9]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v14i1r35\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v14i1r35\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:03:49Z","timestamp":1579320229000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v14i1r35"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,5,9]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2007,1,3]]}},"URL":"https:\/\/doi.org\/10.37236\/953","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2007,5,9]]},"article-number":"R35"}}