{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T16:01:06Z","timestamp":1776441666811,"version":"3.51.2"},"reference-count":5,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":7223,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Networks"],"published-print":{"date-parts":[[1987,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In [G. Gunther, Neighbor\u2010connectivity in regular graphs. <jats:italic>Discrete Appl. Math.<\/jats:italic> <jats:bold>11<\/jats:bold> (1985) 233\u2013243] Gunther introduced the concept of a <jats:italic>k<\/jats:italic> neighbor\u2010connected graph, which has the property that the removal of any <jats:italic>k<\/jats:italic> \u2212 1 closed neighborhoods neither disconnects the graph, nor leaves only a complete graph. In this paper we pursue the investigation of minimal graphs that are <jats:italic>k<\/jats:italic>\u2010regular in addition to being <jats:italic>k<\/jats:italic> neighbor\u2010connected. In a private communication, Gunther conjectured that if <jats:italic>G<\/jats:italic> is such a graph which contains no cliques of size larger than <jats:italic>m<\/jats:italic>, then |V(<jats:italic>G<\/jats:italic>)| \u2267 <jats:italic>k<jats:sup>2<\/jats:sup><\/jats:italic> + (<jats:italic>k<\/jats:italic> + 1 \u2212 <jats:italic>m<\/jats:italic>) (<jats:italic>k<\/jats:italic> \u2212 1) + 1. In the above reference, he proved that this conjecture is valid in the case that <jats:italic>m<\/jats:italic> = <jats:italic>k<\/jats:italic> and characterized the minimal graphs. In this paper, we begin to investigate the case where <jats:italic>m<\/jats:italic> = 2. We give some results connecting the minimal graphs to other combinatorial objects.<\/jats:p>","DOI":"10.1002\/net.3230170208","type":"journal-article","created":{"date-parts":[[2007,5,11]],"date-time":"2007-05-11T22:17:01Z","timestamp":1178921821000},"page":"241-247","source":"Crossref","is-referenced-by-count":17,"title":["Neighbor\u2010connected graphs and projective planes"],"prefix":"10.1002","volume":"17","author":[{"given":"G.","family":"Gunther","sequence":"first","affiliation":[]},{"given":"B. L.","family":"Hartnell","sequence":"additional","affiliation":[]},{"given":"R.","family":"Nowakowski","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,11]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(85)90075-7"},{"key":"e_1_2_1_3_2","unstructured":"G.GuntherandB. L.Hartnell On minimizing the effects of betrayals in a resistance movement. Proc. Eighth Manitoba Conference on Numerical Mathematics and Computing (1978) 285\u2013306."},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(80)90042-6"},{"key":"e_1_2_1_5_2","unstructured":"B. L.Hartnell The optimum defense against random subversions in a network. Proceedings of the Tenth Southeastern conference on Combinatorics Graph Theory and Computing Boca Raton Florida (April1979) 293\u2013299."},{"key":"e_1_2_1_6_2","volume-title":"Projectives Planes","author":"Hughes D. R.","year":"1972"}],"container-title":["Networks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnet.3230170208","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/net.3230170208","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,21]],"date-time":"2023-10-21T16:25:31Z","timestamp":1697905531000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/net.3230170208"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1987,1]]},"references-count":5,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1987,1]]}},"alternative-id":["10.1002\/net.3230170208"],"URL":"https:\/\/doi.org\/10.1002\/net.3230170208","archive":["Portico"],"relation":{},"ISSN":["0028-3045","1097-0037"],"issn-type":[{"value":"0028-3045","type":"print"},{"value":"1097-0037","type":"electronic"}],"subject":[],"published":{"date-parts":[[1987,1]]}}}