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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,43 @@ | ||
| template <int MAXN = 100000, int MAXM = 100000> | ||
| struct chordal_graph { | ||
| int n; edge_list <MAXN, MAXM> e; | ||
| int id[MAXN], seq[MAXN]; | ||
| void init () { | ||
| struct point { | ||
| int lab, u; | ||
| point (int lab = 0, int u = 0) : lab (lab), u (u) {} | ||
| bool operator < (const point &a) const { return lab < a.lab; } }; | ||
| std::fill (id, id + n, -1); | ||
| static int label[MAXN]; std::fill (label, label + n, 0); | ||
| std::priority_queue <point> q; | ||
| for (int i = 0; i < n; ++i) q.push (point (0, i)); | ||
| for (int i = n - 1; i >= 0; --i) { | ||
| for (; ~id[q.top ().u]; ) q.pop (); | ||
| int u = q.top (); q.pop (); id[u] = i; | ||
| for (int j = e.begin[u]; ~j; j = e.next[j]) | ||
| if (v = e.dest[j], !~id[v]) ++label[v], q.push (point (label[v], v)); } | ||
| for (int i = 0; i < n; ++i) seq[id[i]] = i; } | ||
| bool is_chordal () { | ||
| static int vis[MAXN], q[MAXN]; std::fill (vis, vis + n, 0); | ||
| for (int i = n - 1; i >= 0; --i) { | ||
| int u = seq[i], t = 0, v; | ||
| for (int j = e.begin[u]; ~j; j = e.next[j]) | ||
| if (v = e.dest[j], id[v] > id[u]) q[t++] = v; | ||
| if (!t) continue; w = q[0]; | ||
| for (int j = 0; j < t; ++j) if (id[q[j]] < id[w]) w = q[j]; | ||
| for (int j = e.begin[w]; ~j; j = e.next[j]) vis[e.dest[j]] = i; | ||
| for (int j = 0; j < t; ++j) if (q[j] != w && vis[q[j]] != i) return 0; | ||
| } | ||
| return 1; } | ||
| int min_color () { | ||
| int res = 0; | ||
| static int vis[MAXN], c[MAXN]; | ||
| std::fill (vis, vis + n, 0); | ||
| std::fill (c, c + n, n); | ||
| for (int i = n - 1; i >= 0; --i) { | ||
| int u = seq[i]; | ||
| for (int j = e.begin[u]; ~j; j = e.next[j]) vis[c[e.dest[j]]] = i; | ||
| int k; for (k = 0; vis[k] == i; ++k); | ||
| c[u] = k; res = std::max (res, k + 1); | ||
| } | ||
| return res; } }; |
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| A chordal graph is one in which all cycles of four or more vertices have a chord, which is an edge that is not part of the cycle but connects two vertices of the cycle. | ||
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| A perfect elimination ordering in a graph is an ordering of the vertices of the graph such that, for each vertex $v$, $v$ and the neighbors of $v$ that occur after $v$ in the order form a clique. A graph is chordal if and only if it has a perfect elimination ordering. One application of perfect elimination orderings is finding a maximum clique of a chordal graph in polynomial-time, while the same problem for general graphs is NP-complete. More generally, a chordal graph can have only linearly many maximal cliques, while non-chordal graphs may have exponentially many. To list all maximal cliques of a chordal graph, simply find a perfect elimination ordering, form a clique for each vertex $v$ together with the neighbors of $v$ that are later than $v$ in the perfect elimination ordering, and test whether each of the resulting cliques is maximal. | ||
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| The largest maximal clique is a maximum clique, and, as chordal graphs are perfect, the size of this clique equals the chromatic number of the chordal graph. Chordal graphs are perfectly orderable: an optimal coloring may be obtained by applying a greedy coloring algorithm to the vertices in the reverse of a perfect elimination ordering. | ||
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| In any graph, a vertex separator is a set of vertices the removal of which leaves the remaining graph disconnected; a separator is minimal if it has no proper subset that is also a separator. Chordal graphs are graphs in which each minimal separator is a clique. | ||
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| Usage: | ||
| \begin{enumerate} | ||
| \item Set \texttt{n} and \texttt{e}. | ||
| \item Call \texttt{init} to obtain the perfect elimination ordering in \texttt{seq}. | ||
| \item Use \texttt{is\_chordal} to test whether the graph is chordal. | ||
| \item Use \texttt{min\_color} to obtain the size of the maximum clique (and the chromatic number). | ||
| \end{enumerate} | ||
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| \lstinputlisting{src/graph/characteristic/chordal-graph.cpp} |