26 research outputs found

    Determination of the prime bound of a graph

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    Given a graph GG, a subset MM of V(G)V(G) is a module of GG if for each vV(G)Mv\in V(G)\setminus M, vv is adjacent to all the elements of MM or to none of them. For instance, V(G)V(G), \emptyset and {v}\{v\} (vV(G)v\in V(G)) are modules of GG called trivial. Given a graph GG, ωM(G)\omega_M(G) (respectively αM(G)\alpha_M(G)) denotes the largest integer mm such that there is a module MM of GG which is a clique (respectively a stable) set in GG with M=m|M|=m. A graph GG is prime if V(G)4|V(G)|\geq 4 and if all its modules are trivial. The prime bound of GG is the smallest integer p(G)p(G) such that there is a prime graph HH with V(H)V(G)V(H)\supseteq V(G), H[V(G)]=GH[V(G)]=G and V(H)V(G)=p(G)|V(H)\setminus V(G)|=p(G). We establish the following. For every graph GG such that max(αM(G),ωM(G))2\max(\alpha_M(G),\omega_M(G))\geq 2 and log2(max(αM(G),ωM(G)))\log_2(\max(\alpha_M(G),\omega_M(G))) is not an integer, p(G)=log2(max(αM(G),ωM(G)))p(G)=\lceil\log_2(\max(\alpha_M(G),\omega_M(G)))\rceil. Then, we prove that for every graph GG such that max(αM(G),ωM(G))=2k\max(\alpha_M(G),\omega_M(G))=2^k where k1k\geq 1, p(G)=kp(G)=k or k+1k+1. Moreover p(G)=k+1p(G)=k+1 if and only if GG or its complement admits 2k2^k isolated vertices. Lastly, we show that p(G)=1p(G)=1 for every non prime graph GG such that V(G)4|V(G)|\geq 4 and αM(G)=ωM(G)=1\alpha_M(G)=\omega_M(G)=1.Comment: arXiv admin note: text overlap with arXiv:1110.293

    3-uniform hypergraphs: modular decomposition and realization by tournaments

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    Let HH be a 3-uniform hypergraph. A tournament TT defined on V(T)=V(H)V(T)=V(H) is a realization of HH if the edges of HH are exactly the 3-element subsets of V(T)V(T) that induce 3-cycles. We characterize the 3-uniform hypergraphs that admit realizations by using a suitable modular decomposition

    The simplicity index of tournaments

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    An nn-tournament TT with vertex set VV is simple if there is no subset MM of VV such that 2Mn12\leq \left \vert M\right \vert \leq n-1 and for every xVMx\in V\setminus M, either MxM\rightarrow x or xMx \rightarrow M. The simplicity index of an nn-tournament TT is the minimum number s(T)s(T) of arcs whose reversal yields a non-simple tournament. M\"{u}ller and Pelant (1974) proved that s(T)n12s(T)\leq\frac{n-1}{2}, and that equality holds if and only if TT is doubly regular. As doubly regular tournaments exist only if n3(mod4)n\equiv 3\pmod{4}, s(T)<n12s(T)<\frac{n-1}{2} for n≢3(mod4)n\not\equiv3\pmod{4}. In this paper, we study the class of nn-tournaments with maximal simplicity index for n≢3(mod4)n\not\equiv3\pmod{4}
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