Distance measures for embedded graphs

Minimum-Complexity Graph Simplification under Fréchet-Like Distances

“…The traversal distance converts graphs into curves by traversing the graphs continuously and comparing the resulting curves using the Fréchet distance. In other words, it compares the traversal of a man on a graph with the traversal of his dog on part of the other graph while staying close to each other [1].…”

“…This might be helpful in capturing the structure of the graph (see e.g. [1]). See Figure 1 to understand the relationships between the restrictions.…”

“…The former does not respect the continuity of the curves and the latter is also a generalization of the Fréchet distance to graphs, but NP-hard to compute between them. In this paper, we focus on two Fréchet-like distance measures between graphs and/or tree; the graph distance proposed in [1] and traversal distance in [2]. As shown in [1], the traversal distance is not greater than the graph distance.…”

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Simplifying graphs is a very applicable problem in numerous domains especially in computational geometry. Given a geometric graph and a threshold, the minimumcomplexity graph simplification asks for computing an alternative graph of minimum complexity so that the distance between the two graphs remains at most the threshold.In this paper we propose several NP-hardness and algorithmic results depending on the type of input and simplified graphs, the vertex placement of the simplified graph, and the distance measures between them (graph and traversal distances [1,2]). In general, we show that for arbitrary input and output graphs, the problem is NP-hard under some specific vertex-placement of the simplified graph.

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“…In this paper we propose several NP-hardness and algorithmic results depending on the type of input and simplified graphs, the vertex placement of the simplified graph, and the distance measures between them (graph and traversal distances [1,2]). In general, we show that for arbitrary input and output graphs, the problem is NP-hard under some specific vertex-placement of the simplified graph.…”