Abstract
Graph editing problems ask whether an input graph can be modified to a graph with a given property by inserting and deleting vertices and edges. We consider the problem Graph Edit to NDL, which asks whether a graph can be modified to a graph with a given neighbourhood degree list (NDL) using at most \(\ell \) graph edits. The NDL lists the degrees of the neighbours of vertices in a graph, and is a stronger invariant than the degree sequence, which lists the degrees of vertices. In fact, the degree sequence of a graph is determined by its NDL.
We show that Graph Edit to NDL is W[1]-hard when parameterized by \(\ell \) and give an algorithm that runs in fixed-parameter time when parameterized by \(\varDelta +\ell \), where \(\varDelta \) is the maximum degree of the input graph. Furthermore, we adapt our algorithm to solve a harder problem, Constrained Graph Edit to NDL, which imposes constraints on the NDLs of the intermediate graphs produced in the sequence, in fixed-parameter time when parameterized by \(\varDelta +\ell \).
Moreover, there exist graph measures such as assortativity [17] and average nearest neighbour degree [18] that can be derived from the NDL, but not the degree sequence. Our algorithm can be adapted to solve the problem of editing to a graph with a given value of such a measure.
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Nishimura, N., Subramanya, V. (2017). Graph Editing to a Given Neighbourhood Degree List is Fixed-Parameter Tractable. In: Gao, X., Du, H., Han, M. (eds) Combinatorial Optimization and Applications. COCOA 2017. Lecture Notes in Computer Science(), vol 10628. Springer, Cham. https://doi.org/10.1007/978-3-319-71147-8_10
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