Abstract
Given a graph \(G\) and a set \(S\subseteq V(G),\) a vertex \(v\) is said to be \(F_{3}\) -dominated by a vertex \(w\) in \(S\) if either \(v=w,\) or \(v\notin S\) and there exists a vertex \(u\) in \(V(G)-S\) such that \(P:wuv\) is a path in \(G\). A set \(S\subseteq V(G)\) is an \(F_{3}\)-dominating set of \(G\) if every vertex \(v\) is \(F_{3}\)-dominated by a vertex \(w\) in \(S.\) The \(F_{3}\)-domination number of \(G\), denoted by \(\gamma _{F_{3}}(G)\), is the minimum cardinality of an \(F_{3}\)-dominating set of \(G\). In this paper, we study the \(F_{3}\)-domination of Cartesian product of graphs, and give formulas to compute the \(F_{3}\)-domination number of \(P_{m}\times P_{n}\) and \(P_{m}\times C_{n}\) for special \(m,n.\)
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Chang GJ, Chang CW, Kuo D, Poon SH (2012) Algorithmic aspect of stratified domination in graphs, preprint
Chartrand G, Eroh L, Rashidi R, Schultz M, Sherwani NA (1995) Distance, stratified graphs, and greatest stratified subgraphs. Congr Numer 107:81–86
Chartrand G, Gavlas H, Henning MA, Rashidi R (1997) Stratidistance in stratified graphs. Math Bohem 122:337–347
Chartrand G, Haynes TW, Henning MA, Zhang P (2000a) Stratified claw domination in prisms. J Combin Math Combin Comput 33:81–96
Chartrand G, Holley L, Rashidi R, Sherwani NA (2000b) Distance in stratified graphs. Czechoslov Math J 50:35–46
Chartrand G, Haynes TW, Henning MA, Zhang P (2003) Stratification and domination in graphs. Discret Math 272:171–185
Cockayne EJ, Dawes RM, Hedetniemi ST (1980) Total domination in graphs. Networks 10:211–219
Fink JF, Jacobson MS (1985) \(n\)-domination in graphs. In: Alavi Y, Schwenk AJ (eds) Graph theory with applications to algorithms and computer science, (Kalamazoo, MI 1984). Wiley, New York, pp 283–300
Gera R, Zhang P (2004) Bounds for the \(F\)-domination number of a graph. Congr Numer 166:131–144
Gera R, Zhang P (2005a) Realizable triples for stratified domination in graphs. Math Bohem 130:185–202
Gera R, Zhang P (2005b) On stratified domination in oriented graphs. Congr Numer 173:175–192
Gera R, Zhang P (2006) On stratification and domination in graphs. Discuss Math Graph Theory 26:249–272
Gera R, Zhang P (2007) Stratified domination in oriented graphs. J Combin Math Combin Comput 60:105–125
Haynes TW, Hedetniemi ST, Slater PJ (1998a) Fundamentals of domination in graphs. Marcel Dekker, New York
Haynes TW, Hedetniemi ST, Slater PJ (eds) (1998b) Domination in graphs: advanced topics. Marcel Dekker, New York
Haynes TW, Henning MA, Zhang P (2009) A survey of stratified domination in graphs. Discret Math 309:5806–5819
Henning MA, Maritz JE (2004) Stratification and domination in graphs II. Discret Math 286:203–211
Henning MA, Maritz JE (2005) Stratification and domination in graphs with minimum degree two. Discret Math 301:175–194
Henning MA, Maritz JE (2006a) Stratification and domination in prisms. Ars Combin 81:343–358
Henning MA, Maritz JE (2006b) Simultaneous stratification and domination in graphs with minimum degree two. Quaest Math 29:1–16
Rashidi R (1994) The theory and applications of stratified graphs. PhD Dissertation, Western Michigan University, Kalamazoo
Telle JA, Proskurowski A (1997) Algorithms for vertex partitioning problems on partial k-trees. SIAM J Discret Math 10:529–550
Wang FH (2012) An upper bound on \(F\)-domination number in grids, preprint
Acknowledgments
David Kuo was supported in part by the National Science Council undergrants NSC99-2115-M-156-003-MY2. Jing-Ho Yan was supported in part by the National Science Council under grants NSC97-2115-M-259-002-MY3.
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Chang, CW., Kuo, D., Liaw, SC. et al. \(F_{3}\)-domination problem of graphs. J Comb Optim 28, 400–413 (2014). https://doi.org/10.1007/s10878-012-9563-y
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DOI: https://doi.org/10.1007/s10878-012-9563-y