Int.J.ofComputers,Communications&Control,ISSN1841-9836,E-ISSN1841-9844 Vol.III(2008),Suppl.issue:ProceedingsofICCCC2008,pp.508-511 Recognition Algorithm for Antenna-Free Graphs MihaiTalmaciu,ElenaNechita Abstract: During the last three decades, different types of decompositions have been pro- cessed in the field of graph theory. Among these we mention: decompositions based on the additivityofsomecharacteristicsofthegraph,decompositionswheretheadjacencylawbe- tweenthesubsetsofthepartitionisknown,decompositionswherethesubgraphinducedby everysubsetofthepartitionmusthavepredeterminateproperties,aswellascombinationsof suchdecompositions. In various problems in graph theory, for example in the construction of recognition algo- rithms,frequentlyappearstheso-calledweaklydecompositionofgraphs. Inthispaperweintroducethenotionofquasi-weaklydecompositionofagraph G,weshow the existence of a quasi-weakly decomposition, depending on the existence of the weakly decomposition. In addition, we present a recognition algorithm for the class of co-antenna- freegraphs. Keywords: Co-antenna-free graph, weakly decomposition, quasi-weakly decomposition, recognitionalgorithm. 1 Introduction Throughout this paper, G =( V, E ) is a connected, finite and undirected graph, without loops and multiple edges, having V = V (G) asthevertexsetand E = E (G) asthesetofedges. G (or c − G) is the complement of G. If U ⊆ V ,by G( U ) wedenotethesubgraphof G induced by U . By G − X wemeanthesubgraph G( V − X ), whenever X ⊆ V ,butwesimplywrite G − v, when X = {v}. If e = xy isanedgeofagraph G, then x and y are adjacent, while x and e areincident,asare y and e. If xy ∈ E ,wealsouse x ∼ y,and x ∼ y whenever x, y arenot adjacentin G.Avertex z ∈ V distinguishesthenon-adjacentvertices x, y ∈ V if zx ∈ E and zy ∈ E .If A, B ⊂ V are disjointand ab ∈ E forevery a ∈ A and b ∈ B,wesaythat A, B are totally adjacent andwedenoteby A ∼ B,while by A ∼ B wemeanthatnoedgeof G joinssomevertexof A toavertexfrom B and,inthiscase,wesaythat A and B are non-adjacent. The neighbourhood of the vertex v ∈ V istheset N G (v)= {u ∈ V : uv ∈ E }, while N G [v]= N G (v) ∪{v};we simply write N(v) and N[v], when G appears clearly from the context. The neighbourhood of the vertex v inthe complementof G willbedenotedby N(v). If N[v]= V ,then v iscalleda dominating vertex in G. If D ⊂ V andeveryvertexfrom V − D hasatleastone neighbourin D,then D iscalleda dominating set of G.If D ⊂ V and N G (D) = / 0,then D isa non-dominating set of G. Theneighbourhoodof S ⊂ V istheset N(S)= ∪ v∈S N(v) − S and N[S]= S ∪ N(S).A clique isasubset Q of V withthepropertythat G(Q) is complete. The clique number of G,denotedby ω (G),isthesizeofthemaximum clique. By P n , C n , K n wemeanachordlesspathon n ≥ 3vertices,achordlesscycleon n ≥ 3vertices,andacomplete graphon n ≥ 1vertices,respectively. Agraphiscalled triangulated ifitdoesnotcontainchordlesscycleshavingthelengthgreaterorequaltofour. A antenna graphisisomorphicto G =({a, b, c, d .e. f }, {af , fd , fe, db, ec, bc}). Let F denoteafamilyofgraphs.Agraph G iscalled F -free ifnoneofitssubgraphsisin F .The Zykov sum of thegraphs G 1 , G 2 isthegraph G = G 1 + G 2 having: V (G)= V (G 1 ) ∪ V (G 2 ), E (G)= E (G 1 ) ∪ E (G 2 ) ∪{uv : u ∈ V (G 1 ), v ∈ V (G 2 )}. Whensearchingforrecognitionalgorithms,frequentlyappearsatypeofpartitionforthesetofverticesinthree classes A, B, C, which we call a weakly decomposition, such that: A induces a connected subgraph, C is totally adjacentto B,while C and A aretotallynonadjacent. The structure of the paper is the following. In Section 2 we recall the notion of weakly decomposition, and we define the notion a quasi-weakly decomposition. In Section 3 we establish the existence of a quasi-weakly decompositioninagraph G.InSection4wegivearecognitionalgorithmfortheclassofco-antenna-freegraphs. Copyright©2006-2008byCCCPublications-AgoraUniversityEd.House.Allrightsreserved.