A note on theβ-measure for digraph competitions

Digraph games are cooperative TU-games associated to digraph competitions: domination structures that can be modeled by directed graphs. Examples come from sports competitions or from simple majority win digraphs corresponding to preference profiles for a group of individuals within the framework of social choice theory. Brink and Gilles (2000) defined theβ-measure of a digraph competition as the Shapley value of the corresponding digraph game. This paper provides a new characterization of theβ-measure.     

Lightness of digraphs in surfaces and directed game chromatic number

The lightness of a digraph is the minimum arc value, where the value of an arc is the maximum of the in-degrees of its terminal vertices. We determine upper bounds for the lightness of simple digraphs with minimum in-degree at least 1 (resp., graphs with minimum degree at least 2) and a given girth k, and without 4-cycles, which can be embedded in a surface S. (Graphs are considered as digraphs each arc having a parallel arc of opposite direction.) In case k≥5, these bounds are tight for surfaces of nonnegative Euler characteristics. This generalizes results of He et al. [W. He, X. Hou, K.-W. Lih, J. Shao, W. Wang, X. Zhu, Edge-partitions of planar graphs and their game coloring numbers, J. Graph Theory 41 (2002) 307-317] concerning the lightness of planar graphs. From these bounds we obtain directly new bounds for the game colouring number, and thus for the game chromatic number of (di)graphs with girth k and without 4-cycles embeddable in S. The game chromatic resp. game colouring number were introduced by Bodlaender [H.L. Bodlaender, On the complexity of some coloring games, Int. J. Found. Comput. Sci. 2 (1991) 133-147] resp. Zhu [X. Zhu, The game coloring number of planar graphs, J. Combin. Theory B 75 (1999) 245-258] for graphs. We generalize these notions to arbitrary digraphs. We prove that the game colouring number of a directed simple forest is at most 3.     

The Myerson value for directed graph games

A directed graph game consists of a cooperative game with transferable utility and a digraph which describes limited cooperation and the dominance relation among the players. Under the assumption that only coalitions of strongly connected players are able to fully cooperate, we introduce the digraph-restricted game in which a non-strongly connected coalition can only realize the sum of the worths of its strong components. The Myerson value for directed graph games is defined as the Shapley value of the digraph-restricted game. We establish axiomatic characterizations of the Myerson value for directed graph games by strong component efficiency and either fairness or bi-fairness.     

On the core and nucleolus of directed acyclic graph games

We introduce directed acyclic graph (DAG) games, a generalization of standard tree games, to study cost sharing on networks. This structure has not been previously analyzed from a cooperative game theoretic perspective. Every monotonic and subadditive cost game—including monotonic minimum cost spanning tree games—can be modeled as a DAG-game. We provide an efficiently verifiable condition satisfied by a large class of directed acyclic graphs that is sufficient for the balancedness of the associated DAG-game. We introduce a network canonization process and prove various structural results for the core of canonized DAG-games. In particular, we characterize classes of coalitions that have a constant payoff in the core. In addition, we identify a subset of the coalitions that is sufficient to determine the core. This result also guarantees that the nucleolus can be found in polynomial time for a large class of DAG-games.     

A well-quasi-order for tournaments

A digraph H is immersed in a digraph G if the vertices of H are mapped to (distinct) vertices of G, and the edges of H are mapped to directed paths joining the corresponding pairs of vertices of G, in such a way that the paths are pairwise edge-disjoint. For graphs the same relation (using paths instead of directed paths) is a well-quasi-order; that is, in every infinite set of graphs some one of them is immersed in some other. The same is not true for digraphs in general; but we show it is true for tournaments (a tournament is a directed complete graph).     

A solitaire game played on 2-colored graphs

We introduce a solitaire game played on a graph. Initially one disk is placed at each vertex, one face green and the other red, oriented with either color facing up. Each move of the game consists of selecting a vertex whose disk shows green, flipping over the disks at neighboring vertices, and deleting the selected vertex. The game is won if all vertices are eliminated. We derive a simple parity-based necessary condition for winnability of a given game instance. By studying graph operations that construct new graphs from old ones, we obtain broad classes of graphs where this condition also suffices, thus characterizing the winnable games on such graphs. Concerning two familiar (but narrow) classes of graphs, we show that for trees a game is winnable if and only if the number of green vertices is odd, and for n-cubes a game is winnable if and only if the number of green vertices is even and not all vertices have the same color. We provide a linear-time algorithm for deciding winnability for games on maximal outerplanar graphs. We reduce the decision problem for winnability of a game on an arbitrary graph G to winnability of games on its blocks, and to winnability on homeomorphic images of G obtained by contracting edges at 2-valent vertices.     

Locally semicomplete digraphs: A generalization of tournaments

In this paper we introduce a new class of directed graphs called locally semicomplete digraphs. These are defined to be those digraphs for which the following holds: for every vertex x the vertices dominated by x induce a semicomplete digraph and the vertices that dominate x induce a semicomplete digraph. (A digraph is semicomplete if for any two distinct vertices u and ν, there is at least one arc between them.) This class contains the class of semicomplete digraphs, but is much more general. In fact, the class of underlying graphs of the locally semi-complete digraphs is precisely the class of proper circular-arc graphs (see [13], Theorem 3). We show that many of the classic theorems for tournaments have natural analogues for locally semicomplete digraphs. For example, every locally semicomplete digraph has a directed Hamiltonian path and every strong locally semicomplete digraph has a Hamiltonian cycle. We also consider connectivity properties, domination orientability, and algorithmic aspects of locally semicomplete digraphs. Some of the results on connectivity are new, even when restricted to semicomplete digraphs.     

Digraph searching, directed vertex separation and directed pathwidth

The four digraph search models, directed search, undirected search, strong search, and weak search, are studied in this paper. There are three types of actions for searchers in these models: placing, removing, and sliding. The four models differ in the abilities of searchers and intruders depending on whether or not they must obey the edge directions when they move along the directed edges. In this paper, we investigate the relationships between these search models. We introduce the concept of directed vertex separation for digraphs. We also discuss the properties of directed vertex separation, and investigate the relations between directed vertex separation, directed pathwidth and search numbers in different search models.     

Maximum Directed Cuts in Graphs with Degree Constraints

The Max Cut problem is an NP-hard problem and has been studied extensively. Alon et?al. (J Graph Theory 55:1–13, 2007) studied a directed version of the Max Cut problem and observed its connection to the Hall ratio of graphs. They proved, among others, that if an acyclic digraph has m edges and each vertex has indegree or outdegree at most 1, then it has a directed cut of size at least 2m/5. Lehel et?al. (J Graph Theory 61:140–156, 2009) extended this result by replacing the “acyclic digraphs” with the “digraphs containing no directed triangles”. In this paper, we characterize the acyclic digraphs with m edges whose maximum dicuts have exactly 2m/5 edges, and our approach gives an alternative proof of the result of Lehel et?al. We also show that there are infinitely many positive rational numbers β < 2/5 for which there exist digraphs D (with directed triangles) such that each vertex of D has indegree or outdegree at most 1, and any maximum directed cut in D has size precisely β|E(D)|.     

On some balanced, totally balanced and submodular delivery games

This paper studies a class of delivery problems associated with the Chinese postman problem and a corresponding class of delivery games. A delivery problem in this class is determined by a connected graph, a cost function defined on its edges and a special chosen vertex in that graph which will be referred to as the post office. It is assumed that the edges in the graph are owned by different individuals and the delivery game is concerned with the allocation of the traveling costs incurred by the server, who starts at the post office and is expected to traverse all edges in the graph before returning to the post office. A graph G is called Chinese postman-submodular, or, for short, CP-submodular (CP-totally balanced, CP-balanced, respectively) if for each delivery problem in which G is the underlying graph the associated delivery game is submodular (totally balanced, balanced, respectively). For undirected graphs we prove that CP-submodular graphs and CP-totally balanced graphs are weakly cyclic graphs and conversely. An undirected graph is shown to be CP-balanced if and only if it is a weakly Euler graph. For directed graphs, CP-submodular graphs can be characterized by directed weakly cyclic graphs. Further, it is proven that any strongly connected directed graph is CP-balanced. For mixed graphs it is shown that a graph is CP-submodular if and only if it is a mixed weakly cyclic graph. Finally, we note that undirected, directed and mixed weakly cyclic graphs can be recognized in linear time. Received May 20, 1997 / Revised version received August 18, 1998?Published online June 11, 1999     

Competition-reachability of a graph

The reachability r(D) of a directed graph D is the number of ordered pairs of distinct vertices (x,y) with a directed path from x to y. Consider a game associated with a graph G=(V,E) involving two players (maximizer and minimizer) who alternately select edges and orient them. The maximizer attempts to maximize the reachability, while the minimizer attempts to minimize the reachability, of the resulting digraph. If both players play optimally, then the reachability is fixed. Parameters that assign a value to each graph in this manner are called competitive parameters. We determine the competitive-reachability for special classes of graphs and discuss which graphs achieve the minimum and maximum possible values of competitive-reachability.     

Asymmetric directed graph coloring games

This note generalizes the (a,b)-coloring game and the (a,b)-marking game which were introduced by Kierstead [H.A. Kierstead, Asymmetric graph coloring games, J. Graph Theory 48 (2005) 169-185] for undirected graphs to directed graphs. We prove that the (a,b)-chromatic and (a,b)-coloring number for the class of orientations of forests is b+2 if b≤a, and infinity otherwise. From these results we deduce upper bounds for the (a,b)-coloring number of oriented outerplanar graphs and of orientations of graphs embeddable in a surface with bounded girth.     

Game-perfect digraphs

In the A-coloring game, two players, Alice and Bob, color uncolored vertices of a given uncolored digraph D with colors from a given color set C, so that, at any time a vertex is colored, its color has to be different from the colors of its previously colored in-neighbors. Alice begins. The players move alternately, where a move of Bob consists in coloring a vertex, and a move of Alice in coloring a vertex or missing the turn. The game ends when Bob is unable to move. Alice wins if every vertex is colored at the end, otherwise Bob wins. This game is a variant of a graph coloring game proposed by Bodlaender (Int J Found Comput Sci 2:133?C147, 1991). The A-game chromatic number of D is the smallest cardinality of a color set C, so that Alice has a winning strategy for the game played on D with C. A digraph is A-perfect if, for any induced subdigraph H of D, the A-game chromatic number of H equals the size of the largest symmetric clique of H. We characterize some basic classes of A-perfect digraphs, in particular all A-perfect semiorientations of paths and cycles. This gives us, as corollaries, similar results for other games, in particular concerning the digraph version of the usual game chromatic number.     

Irregularity strength of digraphs

It is an elementary exercise to show that any non-trivial simple graph has two vertices with the same degree. This is not the case for digraphs and multigraphs. We consider generating irregular digraphs from arbitrary digraphs by adding multiple arcs. To this end, we define an irregular labeling of a digraph D to be an arc-labeling of the digraph such that the ordered pairs of the sums of the in-labels and out-labels at each vertex are all distinct. We define the strength of D to be the smallest of the maximum labels used across all irregular labelings. Similar definitions for graphs have been studied extensively and a different formulation of digraph irregularity was given in [H. Hackett, Irregularity strength of graphs and digraphs, Masters Thesis, University of Louisville, 1995]. Here we continue the study of irregular labelings of digraphs. We give a general lower bound on and determine exactly for tournaments, directed paths and cycles and the orientation of the path where all vertices have either in-degree 0 or out-degree 0. We also determine the irregularity strength of a union of directed cycles and a union of directed paths, the latter which requires a new result pertaining to finding circuits of given lengths containing prescribed vertices in the complete symmetric digraph with loops.     

Homogeneous games as anti step functions

In this paper the class of homogeneousn-person games “without dummies and steps” is characterized by two algebraic axioms. Each of these games induces a natural vector of lengthn, called incidence vector of the game, and vice versa. A geometrical interpretation of incidence vectors allows to construct all of these games and to enumerate them recursively with respect to the number of persons. In addition an algorithm is defined, which maps each directed game to a minimal representation of a homogeneous game. Moreover both games coincide, if the initial game is homogeneous.     

Sufficient Conditions for Super-Arc-Strongly Connected Oriented Graphs

A digraph D is called super-arc-strongly connected if the arcs of every its minimum arc-disconnected set are incident to or from some vertex in D. A digraph without any directed cycle of length 2 is called an oriented graph. Sufficient conditions for digraphs to be super-arc-strongly connected have been given by several authors. However, closely related conditions for super-arc-strongly connected oriented graphs have little attention until now. In this paper we present some minimum degree and degree sequence conditions for oriented graphs to be super-arc-strongly connected.     

Generalized impartial games

A complete mathematical theory of NIM type games have been developed byBouton [1902],Sprague [1935/36] andGrundy [1939]. The NIM type games are a special class of combinatorial games, called the impartial games. “Impartial” means that, at any stage, the set of legal moves is independent of whose turn it is to move. The outcome of an impartial game is that the first player either wins or loses. The results ofBouton, Sprague andGrundy are now generalized to a wider class of games which allow tie-positions. This wider class of games are defined on digraphs. It is proved that the games defined on a given digraph are all impartial games (without tie-positions) iff the birthday function (also called the terminal distance function) exists on this digraph.     

A characterization of partial directed line graphs

Can a directed graph be completed to a directed line graph? If possible, how many arcs must be added? In this paper we address the above questions characterizing partial directed line (PDL) graphs, i.e., partial subgraph of directed line graphs. We show that for such class of graphs a forbidden configuration criterion and a Krausz's like theorem are equivalent characterizations. Furthermore, the latter leads to a recognition algorithm that requires O(m) worst case time, where m is the number of arcs in the graph. Given a partial line digraph, our characterization allows us to find a minimum completion to a directed line graph within the same time bound.The class of PDL graphs properly contains the class of directed line graphs, characterized in [J. Blazewicz, A. Hertz, D. Kobler, D. de Werra, On some properties of DNA graphs, Discrete Appl. Math. 98(1-2) (1999) 1-19], hence our results generalize those already known for directed line graphs. In the undirected case, we show that finding a minimum line graph edge completion is NP-hard, while the problem of deciding whether or not an undirected graph is a partial graph of a simple line graph is trivial.     

Directed graphical structure,Nash equilibrium,and potential games

This paper considers the directed graphical structure of a game, called influence structure, where a directed edge from player $i$ to player $j$ indicates that player $i$ may be able to affect $j$’s payoff via his unilateral change of strategies. We give a necessary and sufficient condition for the existence of pure-strategy Nash equilibrium of games having a directed graph in terms of the structure of that graph. We also discuss the relationship between the structure of graphs and potential games.     

Maker–Breaker domination game

We introduce the Maker–Breaker domination game, a two player game on a graph. At his turn, the first player, Dominator, selects a vertex in order to dominate the graph while the other player, Staller, forbids a vertex to Dominator in order to prevent him to reach his goal. Both players play alternately without missing their turn. This game is a particular instance of the so-called Maker–Breaker games, that is studied here in a combinatorial context. In this paper, we first prove that deciding the winner of the Maker–Breaker domination game is pspace-complete, even for bipartite graphs and split graphs. It is then showed that the problem is polynomial for cographs and trees. In particular, we define a strategy for Dominator that is derived from a variation of the dominating set problem, called the pairing dominating set problem.