reduction graphs be ARG. It turns out that there are graphs in ARG that are not (isomorphic to) reduction graphs. To obtain a characterization we need one more property of reduction graphs: the ability to linearly order the vertices to resemble its (in general not unique) underlying legal string, as done in Fig. 11. To make this linear order of vertices explicit, we introduce a third set of edges, called merge edges , to the reduction graph as done in Fig. 14. s 2 2 7 7 4 4 7 7 3 3 5 5 3 3 4 4 2 2 6 6 5 5 6 6 t Fig. 14. Merge edges are added to the reduction graph of Fig. 11. Now, when is a set of edges M for G ∈ ARG a set of merge edges? Like desire edges, they have the properties that (1) the edges connect vertices with a common label and (2) each vertex except s and t is connected to exactly one merge edge. Moreover, M and the set E2 are disjoint – no desire edge is parallel to a merge edge. Finally, the reality edges and merge edges must allow for a path from s to t passing each vertex once. This last requirement is equivalent to the fact that the reality and merge edges induce a connected graph. If it is possible to add a set of merge edges to the graph, then it is not difficult to see that the graph is isomorphic to a reduction graph Ru. Indeed, we can identify such a u for this reduction graph by simply considering the alternating path from s to t over the reality and merge edges. The orientation (positiveness or negativeness) of each pointer is determined by the crossing or non crossing of the desire edges (exactly as we defined the notion of reduction graph). To characterize reduction graphs we need the notion of a pointer-component graph. Given an abstract reduction graph, a pointer-component graph describes how the labels of that abstract reduction graph are distributed among its connected components. Definition 5.14. Let G ∈ ARG. The pointer-component graph of G, denoted by PCG, is a multigraph (ζ, E, e), where ζ is the set of connected components of G, E = dom(G) and e is, for e ∈ E, defined by e(e) = {C ∈ ζ | C contains vertices labelled by e}. The pointer-component graph of G = Ru of Fig. 12 is given in Fig. 15. We have ζ = {C1, C2, C3, R} where R is the linear component and the other elements are cyclic components of Ru. It is shown in [7] that, surprisingly, G ∈ ARG has a set of merge edges precisely when the pointer-component graph PCG is a connected graph. In other words: Theorem 5.15 ([7]). An abstract reduction graph G is isomorphic to a reduction graph iff PCG is a connected graph. Computational nature of gene assembly in ciliates 21
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