Minimum gradation in greyscales of graphs
In this paper we present the notion of greyscale of a graph as a colouring of its vertices that uses colours from the real interval [0,1]. Any greyscale induces another colouring by assigning to each edge the non-negative difference between the colours of its vertices. These edge colours are ordered...
| Authors: | , , , |
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| Format: | article |
| Status: | Published version |
| Publication Date: | 2023 |
| Country: | España |
| Institution: | Universidad de Sevilla (US) |
| Repository: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/149060 |
| Online Access: | https://hdl.handle.net/11441/149060 https://doi.org/10.1016/j.disopt.2023.100773 |
| Access Level: | Open access |
| Keyword: | Graph colouring Greyscale Minimum gradation Graph algorithms |
| Summary: | In this paper we present the notion of greyscale of a graph as a colouring of its vertices that uses colours from the real interval [0,1]. Any greyscale induces another colouring by assigning to each edge the non-negative difference between the colours of its vertices. These edge colours are ordered in lexicographical decreasing ordering and give rise to a new element of the graph: the gradation vector. We introduce the notion of minimum gradation vector as a new invariant for the graph and give polynomial algorithms to obtain it. These algorithms also output all greyscales that produce the minimum gradation vector. This way we tackle and solve a novel vectorial optimization problem in graphs that may generate more satisfactory solutions than those generated by known scalar optimization approaches. |
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