{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T20:11:57Z","timestamp":1772136717680,"version":"3.50.1"},"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 $k$-kernel in a digraph $G$ is a stable set $X$ of vertices such that every vertex of $G$ can be joined from $X$ by a directed path of length at most $k$. We prove three results about $k$-kernels.\r\nFirst, it was conjectured by Erd\u0151s and Sz\u00e9kely in 1976 that every digraph $G$ with no source has a 2-kernel $|K|$ with $|K|\\le |G|\/2$. We prove this conjecture when $G$ is a \"split digraph\" (that is, its vertex set can be partitioned into a tournament and a stable set), improving a result of Langlois et al., who proved that every split digraph $G$ with no source has a 2-kernel of size at most $2|G|\/3$.\r\nSecond, the Erd\u0151s-Sz\u00e9kely conjecture implies that in every digraph $G$ there is a 2-kernel $K$ such that the union of $K$ and its out-neighbours has size at least $|G|\/2$. We prove that this is true if $V(G)$ can be partitioned into a tournament and an acyclic set.\r\nThird, in a recent paper, Spiro asked whether, for all $k\\ge 3$, every strongly-connected digraph $G$ has a $k$-kernel of size at most about $|G|\/(k+1)$. This remains open, but we prove that there is one of size at most about $|G|\/(k-1)$.<\/jats:p>","DOI":"10.37236\/13556","type":"journal-article","created":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T19:43:53Z","timestamp":1772135033000},"source":"Crossref","is-referenced-by-count":0,"title":["Distant Digraph Domination"],"prefix":"10.37236","volume":"33","author":[{"given":"Tung","family":"Nguyen","sequence":"first","affiliation":[]},{"given":"Alexander","family":"Scott","sequence":"additional","affiliation":[]},{"given":"Paul","family":"Seymour","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2026,2,13]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p32\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p32\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T19:43:54Z","timestamp":1772135034000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v33i1p32"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,13]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/13556","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,13]]},"article-number":"P1.32"}}