Fractional Maker-Breaker Resolving Game Chapter uri icon

abstract

  • Let G be a graph with vertex set V(G), and let d(u,w) denote the length of a $$u-w$$u-wgeodesic in G. For two distinct $$x,y in V(G)$$x,yV(G), let $$R{x,y}={z in V(G): d(x,z)
    e d(y,z)}$$R{x,y}={zV(G):d(x,z)d(y,z)}. For a function g defined on V(G) and for $$U subseteq V(G)$$UV(G), let $$g(U)=sum _{sin U}g(s)$$g(U)=sUg(s). A real-valued function $$g: V(G)
    ightarrow [0,1]$$g:V(G)[0,1]is a resolving function of G if $$g(R{x,y}) ge 1$$g(R{x,y})1for any distinct $$x,y in V(G)$$x,yV(G). In this paper, we introduce the fractional Maker-Breaker resolving game (FMBRG). The game is played on a graph G by Resolver and Spoiler (denoted by $$R^*$$Rand $$S^*$$S, respectively) who alternately assigns non-negative real values on V(G) such that its sum is at most one on each turn. Moreover, the total value assigned, by $$R^*$$Rand $$S^*$$S, on each vertex over time cannot exceed one. $$R^*$$Rwins if the total values assigned on V(G) by $$R^*$$R, after finitely many turns, form a resolving function of G, whereas $$S^*$$Swins if $$R^*$$Rfails to assign values on V(G) to form a resolving function of G. We obtain some general results on the outcome of the FMBRG and determine the outcome of the FMBRG for some graph classes.

author list (cited authors)

  • Yi, E.

citation count

  • 0

complete list of authors

  • Yi, Eunjeong

editor list (cited editors)

  • Wu, W., & Zhang, Z.

Book Title

  • Combinatorial Optimization and Applications

publication date

  • January 2020