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Happy Cliques


\begin{definition}{Let $G(N,A)$\space be an interval overlap graph. A {\em happy... ...ot\in C$\space where $(z,y)\in A$\space and $(z,x) \not\in A$ } \end{definition}
For example, in the overlap graph shown in figure 10.10(c) $\{(2,3),(10,11)\}$ and $\{(6,7)\}$ are happy cliques, but $\{(2,3),(10,11),(8,9)\}$ is not.


 \begin{claim} The reversal defined by a node $x$\space with maximum unoriented d... ...s) in a happy clique $C$\space creates no new unoriented components. \end{claim}

\begin{proof}Suppose that such a reversal created an unoriented component $M$ . ... ...eighbors then $x$ , a contradiction to the choice of $x$ . \end{list}\end{proof}

Claim 10.11 implies, for example, that the reversal defined by the gray edge (10,11) is a safe proper reversal for the permutation of figure 10.10 (a), since it corresponds to the node with maximum unoriented degree in the happy clique $\{(2,3),(10,11)\}$. On the other hand, the reversal defined by (2,3) creates a new unoriented component, as it yields the permutation shown in figure 10.11.


Peer Itsik
2001-01-17