Alliance Partitions in Graphs.

• Main Author: Lachniet, Jason
• Format: Others
• Published: Digital Commons @ East Tennessee State University 2007
• Subjects: domination; alliance partition; defensive alliance; graph theory; Discrete Mathematics and Combinatorics; Mathematics; Physical Sciences and Mathematics
• Online Access: https://dc.etsu.edu/etd/2080; https://dc.etsu.edu/cgi/viewcontent.cgi?article=3441&context=etd

For a graph G=(V,E), a nonempty subset S contained in V is called a defensive alliance if for each v in S, there are at least as many vertices from the closed neighborhood of v in S as in V-S. If there are strictly more vertices from the closed neighborhood of v in S as in V-S, then S is a strong defensive alliance. A (strong) defensive alliance is called global if it is also a dominating set of G. The alliance partition number (respectively, strong alliance partition number) is the maximum cardinality of a partition of V into defensive alliances (respectively, strong defensive alliances). The global (strong) alliance partition number is defined similarly. For each parameter we give both general bounds and exact values. Our major results include exact values for the alliance partition number of grid graphs and for the global alliance partition number of caterpillars.