Happy Cliques

Let C be a clique of oriented vertices in $OV(\pi )$. We say that C is happy if for every oriented vertex $e\not\in C$ and every vertex $f\in C$ such that $(e,f)\in E(OV(\pi))$ there exists an oriented vertex $g\not\in C$ such that $(g,e)\in E(OV(\pi))$ and $(g,f)\not\in E(OV(\pi))$. For example, in the overlap graph shown in figure 11.9(c) $\{(2,3),(10,11)\}$ and $\{(6,7)\}$ are happy cliques, but $\{(2,3),(10,11),(8,9)\}$ is not. We use the following claim:

Claim 11.13   The reversal defined by a vertex x with maximum unoriented degree (maximum number of unoriented neighbors) in a happy clique C creates no new unoriented components.

Proof:Suppose that such a reversal created an unoriented component M.

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M contains a neighbor y of x:
Suppose otherwise. But we know that M is unoriented. Therefore neighborhood relationships and orientation in M are unchanged, thus M must have been unoriented before the reversal, and this contradicts to the happy clique definition.

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M contains no neighbor of x outside C. Therefore $y\in C$:
Suppose to the contrary that there exist $e \in M(C)$ such that $(e,x) \in E(OV(\pi))$. There are two cases to examine: Either e was unoriented before applying the reversal r, hence e is oriented and so is M - a contradiction. Otherwise, e is oriented, and by the definition of the happy clique C, e has an oriented neighbor g, unadjacent to x. Therefore $g \in M$, and its orientation remains unchanged by applying r, thus M is oriented - a contradiction.

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Every unoriented neighbor of x is adjacent to y:
Suppose to the contrary that z is an unoriented neighbor of x, unadjacent to y. Then after applying r, z is oriented, and adjacent to y, hence $z \in M$, contradicting M being unoriented.

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|M|>1:
Every unoriented edge $\pi_{2i},\pi_{2j-1}$ has a neighbor. Otherwise, suppose i<j and $\pi_{2i}$ is odd (the other cases are analogous). Then $\pi_{2i}+2$ appears between $\pi_{2i}$ and $\pi_{2j-1}$, and so is $\pi_{2i}+2k$ for all k, by induction - a contradiction. Therefore, y has, after applying r, an unoriented neighbor z. Then $z \notin C$ and z is not adjacent to x. Then y has more unoriented neighbors then x, a contradiction to the choice of x.

Claim 11.13 implies,For example, that the reversal defined by the gray edge (10,11) is a safe proper reversal for the permutation of figure 11.9 (a), since it corresponds to the vertex 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 11.10.