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Using the complete biorientation of a mixed graph M , one can easily give the definitions of a walk, trail, path, and cycle ↔ in M . The only extra condition is that every pair of arcs in M obtained in replacement of an edge in M has to be treated as two appearances of one thing. For example, if one of the arcs in such a pair appears in a trail T , then ↔ the second one cannot be in T . A mixed graph M is strong if M is strong. Similarly, one can give the definition of strong components. A mixed graph ↔ M is connected if M is connected.

A mixed graph M is strong if M is strong. Similarly, one can give the definition of strong components. A mixed graph ↔ M is connected if M is connected. An edge in a connected mixed graph M is a bridge if M − is not connected. 14 illustrates the notion of a mixed graph. 14 is strong; u, (u, v), v, {v, u}, u is a cycle in M ; M − x has two strong components: one consists of the vertex y, the other is M = M {u, v, w} ; the edge {v, w} is a bridge in M , the arc (u, v) and the edge {u, v} are not bridges in M ; M is bridgeless.

Let i := i + 1 and go to Step 2. If yi−1 →xi then P := P xi . Let i := i + 1 and go to Step 2. For j = 1 to i − 2 do: If yj →xi →yj+1 then P := P [y1 , yj ]xi P [yj+1 , yi−1 ]. Let i := i + 1 and go to Step 2. 7. Return the path P . The correctness of this algorithm follows from the above proof. To see that this algorithm can be implemented as an O(n2 ) algorithm, observe that the algorithm has two nested loops, each of which perform O(n) operations (we count queries to the adjacency matrix as constant time) and all other steps take constant time.

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Algorithm for factorization of logic expressions by Fallah F


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