The maximum infection time in the geodesic and monophonic convexities

详细信息    查看全文

• 作者：Fabrí ; cio Benevides^a ; ¹ ; ^{fabricio@mat.ufc.br" class="auth_mail" title="E-mail the corresponding author} ; Victor Campos^a ; ¹ ; ^{campos@lia.ufc.br" class="auth_mail" title="E-mail the corresponding author} ; Mitre C. Dourado^b ; ¹ ; ^{mitre@dcc.ufrj.br" class="auth_mail" title="E-mail the corresponding author} ; Rudini M. Sampaio^a ; ¹ ; ^{rudini@lia.ufc.br" class="auth_mail" title="E-mail the corresponding author} ; Ana Silva^a ; ¹ ; ^{anasilva@mat.ufc.br" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Geodesic convexity ; Maximum infection time ; Monophonic hull number ; Monophonic convexity
• 刊名：Theoretical Computer Science
• 出版年：2016
• 出版时间：4 January 2016
• 年：2016
• 卷：609
• 期：part_P2
• 页码：287-295
• 全文大小：373 K

文摘

Recent papers investigated the maximum infection time t_P3(G)$t_{P3}(G)$ of the P₃$P_3$-convexity (also called maximum time of 2-neighbor bootstrap percolation) and the maximum infection time t_mo(G)$t_{mo}(G)$ of the monophonic convexity. In 2014, it was proved that, for bipartite graphs, deciding whether t_P3(G)≥k$t_{P3}(G)\ge k$ is polynomial time solvable for k≤4$k\le 4$, but is NP-complete for k≥5$k\ge 5$[23]. In 2015, it was proved that deciding whether t_mo(G)≥2$t_{mo}(G)\ge 2$ is NP-complete even for bipartite graphs [12]. In this paper, we investigate the maximum infection time cffe" title="Click to view the MathML source">t_gd(G)$t_{gd}(G)$ of the geodesic convexity. We prove that deciding whether t_gd(G)≥k$t_{gd}(G)\ge k$ is polynomial time solvable for k=1$k=1$, but is NP-complete for k≥2$k\ge 2$ even for bipartite graphs. We also present an O(n³m)$O(n^{3}m)$-time algorithm to determine cffe" title="Click to view the MathML source">t_gd(G)$t_{gd}(G)$ and t_mo(G)$t_{mo}(G)$ in distance-hereditary graphs. For this, we characterize all minimal hull sets of a general graph in the monophonic convexity. Moreover, we improve the complexity of the fastest known algorithm for finding a minimum hull set of a general graph in the monophonic convexity.