AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 77(2) (2020), Pages 144–156 Closed-neighborhood union-closed graphs Kelly Guest Tuskegee University Tuskegee, AL, U.S.A. Sarah Holliday Southern Polytechnic State University Marietta, GA, U.S.A. Peter Johnson Auburn University Auburn, AL, U.S.A. Dieter Rautenbach Universit¨ at Ulm Ulm, Baden-W¨ urttemberg, Germany Matthew Walsh Purdue University at Fort Wayne Fort Wayne, IN, U.S.A. Abstract A CNUC graph is one in which the union of any two closed neighborhoods of single vertices is a closed neighborhood of a single vertex. We show that every (finite, simple) graph is an induced subgraph of a CNUC graph. This paper provides the definition of the cnuc number of a graph, the minimum number of vertices that must be added in order to embed the graph as an induced subgraph of a CNUC graph. This number is determined or estimated for several classes of graphs. 1 Introduction All graphs in this paper will be finite and simple. Our notation will be conventional: a graph G has vertex set V (G) and edge set E(G); an edge with end-vertices u, v may be denoted uv or vu; if v ∈ V (G), N G (v)= {u ∈ V (G) | uv ∈ E(G)}, the open ISSN: 2202-3518 c The author(s). Released under the CC BY-ND 4.0 International License