Reachability in Graph Transformation Systems and Slice Languages

Abstract

In this work we show that the reachability problem for graph transformation systems is in the complexity class XP when parameterized with respect to the depth of derivations and the cutwidth of the source graph. More precisely, we show that for any set $\mathcal {R}$ of graph transformation rules, one can determine in time $f(c,d)\cdot |G|\cdot |H|^{g(c,d)}$ whether a graph G of cutwidth c can be transformed into a graph H in depth at most d by the application of graph transformation rules from $\mathcal {R}$. In particular, our algorithm runs in polynomial time when c and d are constants. On the other hand, we show that the problem becomes NP-hard if we allow $c=O(|G|)$ and $d=5$. In the case in which all transformation rules are monotone we get an algorithm running in time $f(c,d)\cdot |G|^{O(c)}\cdot |H|$. To prove our main theorems we will establish an interesting connection between graph transformation systems and regular slice languages. More precisely, we show that if $\mathcal {A}$ is a slice automaton representing a set ${\mathcal {L}}_{{\mathcal {G}}}(\mathcal {A})$ of graphs, then one can construct in time linear in $|\mathcal {A}|$ a slice automaton $\mathcal {N}(\mathcal {A})$ representing the set of all graphs that can be obtained from graphs in ${\mathcal {L}}_{{\mathcal {G}}}(\mathcal {A})$ by the application of one layer of transformation rules in $\mathcal {R}$.