$H$-Kernels by walks in subdivision digraph

Document Type : Research Paper


1 Ciudad Universitaria,Coyoacán 04510,Ciudad de México, México

2 Universidad Autónoma del Estado de México, Estado de México

3 Department of Mathematics, Science Faculty, UNAM


Let $H$ be a digraph possibly with loops and $D$ a digraph without loops whose arcs are colored with the vertices of $H$ ($D$ is said to be an $H$-colored digraph)‎. ‎A directed walk $W$ in $D$ is said to be an $H$-walk if and only if the consecutive colors encountered on $W$ form a directed walk in $H$‎. ‎A subset $N$ of the vertices of $D$ is said to be an $H$-kernel by walks if (1) for every pair of different vertices in $N$ there is no $H$-walk between them ($N$ is $H$-independent by walks) and (2) for each vertex $u$ in $V$($D$)-$N$ there exists an $H$-walk from $u$ to $N$ in $D$ ($N$ is $H$-absorbent by walks)‎. ‎Suppose that $D$ is a digraph possibly infinite‎. ‎In this paper we will work with the subdivision digraph $S_H$($D$) of $D$‎, ‎where $S_H$($D$) is an $H$-colored digraph defined as follows‎: ‎$V$($S_H$($D$)) = $V$($D$) $\cup$ $A$($D$) and $A$($S_H$($D$)) = \{($u$,$a$)‎ : ‎$a$ = ($u$,$v$) $\in$ $A$($D$)\} $\cup$ \{($a$,$v$)‎ : ‎$a$ = ($u$,$v$) $\in$ $A$($D$)\}‎, ‎where ($u$‎, ‎$a$‎, ‎$v$) is an $H$-walk in $S_H$($D$) for every $a$ = ($u$,$v$) in $A$($D$)‎. ‎We will show sufficient conditions on $D$ and on $S_H$($D$) which guarantee the existence or uniqueness of $H$-kernels by walks in $S_H$($D$)‎.


[1] P. Arpin and V. Linek, Reachability problems in edge-colored digraphs, Discrete Math., 307 (2007) 2276–2289.
[2] J. Bang-Jensen and G. Gutin, Digraphs: Theory, Algorithms and Applications, Springer, London, 2000.
[3] C. Berge, Graphs, North-Holland, Amsterdan, 1989.
[4] E. Boros, V. Gurvich, Perfect graphs, kernels and cores of cooperative games, RUTCOR Research Report 12.
Rutgers University, april 2003.
[5] V. Chvátal, On the computational complexity of finding a kernel, Report CRM300, Centre de Recherches
Mathématiques, Université de Montréal, 1973.
[6] A. S. Fraenkel, Combinatorial Games: Selected Bibliography with a Succinct Gourmet Introduction, Electron J.
Combin., 14 (DS2) (2009).
[7] H. Galeana-Sánchez and R. Rojas-Monroy, Kernels and some operations in edge-coloured digraphs, Discrete Math.,
308 (2008) 6036–6046.
[8] T. W. Haynes, S.T. Hedetniemi and P. J. Slater, Domination in graphs: advanced topics. Monographs and textbooks
in pure and applied mathematics 209, Marcel Dekker Inc, New York, 1998.
[9] T. W. Haynes, S.T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs, Chapman and Hall/CRC
Pure and Applied Mathematics 464. Marcel Dekker Inc, New York, 1998.
[10] V. Linek and B. Sands, A note on paths in edge-colored tournaments, Ars Combin., 44 (1996) 225–228.
[11] von J. Neumann and O. Morgenstern, Theory of games and economic behavior, Princeton University Press, Prince-
ton, NJ, 1944.
[12] R. Rojas-Monroy and J. I. Villarreal-Valdés, Kernels in infinite digraphs, AKCE J. Graphs. Combin., 7 no. 1 (2010)
[13] B. Sands, N. Sauer and R. Woodrow, On monochromatic paths in edge coloured digraphs, J. Combin. Theory Ser.
B, 33 (1982) 271–275.
[14] J. Topp, kernels of digraphs formed by some unary operations from other digraphs, J. Rostock Math. Kolloq., 21
(1982) 73–81.
  • Receive Date: 10 November 2017
  • Revise Date: 15 February 2019
  • Accept Date: 05 January 2020
  • Published Online: 01 June 2020