Even Kernels

10.37236/1185 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Aviezri Fraenkel
Keyword(s):  
Game Theory ◽  
Bipartite Graph ◽  
Set Theory ◽  
Combinatorial Game Theory ◽  
Independent Subset ◽  
Combinatorial Game ◽  
Np Complete

Given a graph $G = (V,E)$, an even kernel is a nonempty independent subset $V' \subseteq V$, such that every vertex of $G$ is adjacent to an even number (possibly 0) of vertices in $V'$. It is proved that the question of whether a graph has an even kernel is NP-complete. The motivation stems from combinatorial game theory. It is known that this question is polynomial if $G$ is bipartite. We also prove that the question of whether there is an even kernel whose size is between two given bounds, in a given bipartite graph, is NP-complete. This result has applications in coding and set theory.

scholarly journals Combinatorial Game Theory Foundations Applied to Digraph Kernels

10.37236/1325 ◽  
1996 ◽  
Vol 4 (2) ◽  
Author(s):  
Aviezri S. Fraenkel
Keyword(s):  
Game Theory ◽  
Opposite Direction ◽  
Decision Problem ◽  
Independent Interest ◽  
Combinatorial Game Theory ◽  
Restricted Class ◽  
Combinatorial Game ◽  
Induced Subgraphs ◽  
Vertex Set ◽  
Np Complete

Known complexity facts: the decision problem of the existence of a kernel in a digraph $G=(V,E)$ is NP-complete; if all of the cycles of $G$ have even length, then $G$ has a kernel; and the question of the number of kernels is $\#$P-complete even for this restricted class of digraphs. In the opposite direction, we construct game theory tools, of independent interest, concerning strategies in the presence of draw positions, to show how to partition $V$, in $O(|E|)$ time, into $3$ subsets $S_1,S_2,S_3$, such that $S_1$ lies in all the kernels; $S_2$ lies in the complements of all the kernels; and on $S_3$ the kernels may be nonunique. Thus, in particular, digraphs with a "large" number of kernels are those in which $S_3$ is "large"; possibly $S_1=S_2=\emptyset$. We also show that $G$ can be decomposed, in $O(|E|)$ time, into two induced subgraphs $G_1$, with vertex-set $S_1\cup S_2$, which has a unique kernel; and $G_2$, with vertex-set $S_3$, such that any kernel $K$ of $G$ is the union of the kernel of $G_1$ and a kernel of $G_2$. In particular, $G$ has no kernel if and only if $G_2$ has none. Our results hold even for some classes of infinite digraphs.

scholarly journals Special issue on combinatorial game theory

2018 ◽  
Vol 47 (2) ◽  
pp. 375-377
Author(s):  
Aviezri Fraenkel ◽  
Urban Larsson ◽  
Carlos P. Santos ◽  
Bernhard von Stengel
Keyword(s):  
Game Theory ◽  
Combinatorial Game Theory ◽  
Special Issue ◽  
Combinatorial Game

scholarly journals On lattices from combinatorial game theory modularity and a representation theorem: Finite case

2014 ◽  
Vol 527 ◽  
pp. 37-49 ◽  
Author(s):  
Alda Carvalho ◽  
Carlos Pereira dos Santos ◽  
Cátia Dias ◽  
Francisco Coelho ◽  
João Pedro Neto ◽  
...  
Keyword(s):  
Game Theory ◽  
Representation Theorem ◽  
Combinatorial Game Theory ◽  
Combinatorial Game ◽  
Finite Case

scholarly journals New results for Domineering from combinatorial game theory endgame databases

2015 ◽  
Vol 592 ◽  
pp. 72-86 ◽  
Author(s):  
Jos W.H.M. Uiterwijk ◽  
Michael Barton
Keyword(s):  
Game Theory ◽  
Combinatorial Game Theory ◽  
Combinatorial Game
Sign in / Sign up

Export Citation Format

Share Document