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A crown in a graph $G$ is a pair $(H, C)$, where $H \subseteq V(G)$ and $C \subseteq V(G)$ with $H ∩ C = ∅$ such that the following conditions hold:

(a) The set of neighbors of vertices in $C$ is precisely $H$, i.e. $H = N(C)$,

(b) $C = C_m \cup C_u$ is an independent set, and

(c) There is a perfect matching between $C_m$ and $H$

Theorem:

Any graph $G(V,E)$ with an independent set $I$, where $|I| ≥ \dfrac{2n}{3}$ , has a crown $(H, C)$, where $H ⊆ N(I), C ⊆ I$ and $C_u \ne ∅$, that can be found in time $O(nm)$ given $I$. $(n=|V|, m = |E|)$


I am trying to code an algorithm to find such a crown in $O(nm)$ time

The only way I could think of finding a crown given an independent set is to take subsets of the set: $I$ and check for all the three conditions of a crown. However, taking all subsets of $I$ will take exponential time right? How to find it in $O(nm)$ time. I just want an algorithm with polynomial time. It need not be $O(nm)$... It can be in the form $O(n^c m^d)$ where $c,d$ are constants.

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In the book Kernelization by Fomin, Lokshtanov, Saurabh, Zehavi (which is downloadable as a PDF on the first author's homepage) you find Chapter 4. Crown decomposition. This should point you to the answer.

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  • $\begingroup$ I didnt find the answer. I searched that chapter $\endgroup$
    – Sid
    May 7 at 4:10
  • $\begingroup$ Please specify the page number $\endgroup$
    – Sid
    May 7 at 4:10
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    $\begingroup$ Where in the algorithm are you stuck? $\endgroup$
    – Pål GD
    May 7 at 7:43

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