Abstract
It is \(\textsf {NP}\)-hard to determine the minimum number of branching vertices needed in a single-source distance-preserving subgraph of an undirected graph. We show that this problem can be solved in polynomial time if the input graph is an interval graph.
In earlier work, it was shown that every interval graph with k terminal vertices admits an all-pairs distance-preserving subgraph with \(O(k\log k)\) branching vertices [13]. We extend this result to bi-interval graphs; these are graphs that can be expressed as the strong product of two interval graphs. We present a polynomial time algorithm that takes a bi-interval graph with k terminal vertices as input, and outputs an all-pairs distance-preserving subgraph of it with \(O(k^2)\) branching vertices. This bound is tight.
K. Gajjar—Supported by a DAE scholarship.
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Notes
- 1.
The container cannot be left at the warehouse/storage unit of port H itself beyond a certain limited period of time.
- 2.
The boxicity of a graph is the minimum dimension in which a given graph can be represented as an intersection graph of axis-parallel boxes.
- 3.
Every interval graph is a bi-interval graph.
- 4.
Note that when restricted to a fixed row (or a fixed column), a bi-interval graph is simply an interval graph.
- 5.
Anti-parallel paths are two greedy shortest paths in opposite directions. We do not consider southeast or northwest paths because southeast is anti-parallel to northwest, and southwest is anti-parallel to northeast, and there does not seem to be any straightforward method to get a handle on the number of branching vertices in terms of k by using anti-parallel paths (see [14, Section 3.7] for a more thorough explanation of this).
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Gajjar, K., Radhakrishnan, J. (2019). Minimizing Branching Vertices in Distance-Preserving Subgraphs. In: van Bevern, R., Kucherov, G. (eds) Computer Science – Theory and Applications. CSR 2019. Lecture Notes in Computer Science(), vol 11532. Springer, Cham. https://doi.org/10.1007/978-3-030-19955-5_12
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