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The concept of rainbow connection number of a graph was introduced by Chartrand et al. in 2008. Inspired by this concept, other concepts on colored version of connectivity in graphs were introduced, such as the monochromatic connection number by Caro and Yuster in 2011, the proper connection number by Borozan et al. in 2012, and the conflict-free connection number by Czap et al. in 2018, as well as some other variants of connection numbers later on. Chakraborty et al. proved that to compute the rainbow connection number of a graph is NP-hard. For a long time, it has been tried to fix the computational complexity for the monochromatic connection number, the proper connection number and the conflict-free connection number of a graph. However, it has not been solved yet. Only the complexity results for the strong version, i.e., the strong proper connection number and the strong conflict-free connection number, of these connection numbers were determined to be NP-hard. In this paper, we prove that to compute each of the monochromatic connection number, the proper connection number and the conflict free connection number for a graph is NP-hard. This solves a long standing problem in this field, asked in many talks of workshops and papers. [ABSTRACT FROM AUTHOR]

This paper presents a new algorithmic framework for some basic problems on binary images. Algorithms for binary images such as one of extracting a connected component containing a query pixel and that of connected components labeling play basic roles in image processing. Those algorithms usually use linear work space for efficient implementation. In this paper we propose algorithms for several basic problems on binary images which are efficient in time and space, using space-efficient algorithms for grid graphs. More exactly, some of them run in O ( n log n ) time using O ( 1 ) work space and the others run in O ( n ) or O ( n log n ) time using O ( n ) work space for a binary image of n pixels stored in a read-only array. [ABSTRACT FROM AUTHOR]

The concept of rainbow connection was introduced by Chartrand et al. [] in 2008. It is interesting and recently quite a lot papers have been published about it. In this survey we attempt to bring together most of the results and papers that dealt with it. We begin with an introduction, and then try to organize the work into five categories, including (strong) rainbow connection number, rainbow k-connectivity, k-rainbow index, rainbow vertex-connection number, algorithms and computational complexity. This survey also contains some conjectures, open problems and questions. [ABSTRACT FROM AUTHOR]