Oriented graphs

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χ-bounded families of oriented graphs

χ-bounded families of oriented graphs

Abstract A famous conjecture of Gy´arf´as and Sumner states for any tree T and integer k, if the chromatic number of a graph is large enough, either the graph contains a clique of size k or it contains T as an induced subgraph. We discuss some results and open problems about extensions of this conjecture to oriented graphs. We conjecture that for every oriented star S and integer k, if the chromatic number of a digraph is large enough, either the digraph contains a clique of size k or it contains S as an induced subgraph. As an evidence, we prove that for any oriented star S, every oriented graph with sufficiently large chromatic number contains either a transitive tournament of order 3 or S as an induced subdigraph. We then study for which sets P of orientations of P 4 (the path on four vertices) similar statements
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$χ$-bounded families of oriented graphs

$χ$-bounded families of oriented graphs

either semicomplete, or semicomplete bipartite, or in the set F of oriented graphs D that have three vertices {v 1 , v 2 , v 3 } such that A(D) = {v 1 v 2 , v 2 v 3 , v 3 v 1 } ∪ S u∈V (D)\{v 1 ,v 2 ,v 3 } {v 1 u, uv 2 }. Recall that a digraph D is semicomplete if for any two vertices u, v ∈ V (D) at least one of the two arcs uv and vu is in A(D), and that it is semicomplete bipartite if there is a bipartition (A, B) of V (D) such that if for any a ∈ A and b ∈ B, at least one of the two arcs ab and ba is in A(D). Since semicomplete digraphs and members of F are not (~ C 3 , T T 3 )-free, strong (~ C 3 , T T 3 , P + (3))-free oriented graphs are bipartite tournaments and consequently have chromatic number at most 2. On the other hand, Forb(P + (3)) ∩ S is not χ-bounded. Indeed, adding to every acyclic (T T 3 , P + (3))-free oriented graph D a vertex x which dominates all sources of D and is dominated by all other vertices, we obtain a strong (Or(K 4 ), P + (3))-free oriented graph D 0 with chromatic number χ(D) + 1; since χ(Forb(~ C , T T 3 , P + (3))) = +∞, we get χ(Forb(Or(K4 ), P + (3)) ∩ S ) = +∞.
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Metric Dimension: from Graphs to Oriented Graphs

Metric Dimension: from Graphs to Oriented Graphs

later exhibited (Cayley digraphs [5], line digraphs [6], tournaments [9], digraphs with cyclic covering [11], De Bruijn and Kautz digraphs [12], etc.). 1.3 From undirected graphs to oriented graphs To avoid any confusion, let us recall that an orientation D of an undirected graph G is obtained when every edge uv of G is oriented either from u to v (resulting in the arc (u, v)) or conversely (resulting in the arc (v, u)). An oriented graph D is a directed graph that is an orientation of a simple graph. Note that when G is simple, D cannot have two vertices u, v such that (u, v) and ( v, u) are arcs. Such symmetric arcs are allowed in digraphs, which is the main difference between oriented graphs and digraphs. Throughout this paper, when simply referring to a graph, we mean an undirected graph.
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Percolation and first-passage percolation on oriented graphs

Percolation and first-passage percolation on oriented graphs

deterministic cone gives the directions in which infinite paths are found. Between these two models, it seems natural to ask what may happen for per- colation for oriented graphs, on the set of vertices Z d , whose connections do not forbid any direction, or in other words, for oriented graphs that contain loops. In the present paper, we first exhibit one example of such an oriented graph, where every direction is permitted, but such that we observe two phase transitions: if p is small, there there exists no infinite path, then when p increases there is a phase where infinite paths exist but not in any direction (as in classical supercritical oriented percolation), and finally, when p is large enough, infinite paths can grow in any direction (as in classical supercritical unoriented percolation).
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On bialgebras and Hopf algebras of oriented graphs

On bialgebras and Hopf algebras of oriented graphs

where I(Γ) is the number of internal edges of the graph Γ and where V (Γ) is the number of vertices. We shall mainly focus on cycle-free oriented graphs, for which there exists a poset structure on the set of vertices: namely, v < w if and only if there exists a path from v to w, i.e. a collection (e 1 , . . . , e n ) of edges such that the target of e k coincides with the source of e k+1 for k = 1, . . . , n − 1, and such that v (resp. w) is the source

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Metric Dimension: from Graphs to Oriented Graphs

Metric Dimension: from Graphs to Oriented Graphs

however, the orientation D of P obtained by making every vertex v 2k+1 (k = 0, ..., n − 1) become a source (i.e., orienting its incident edges away) veries MD(D) = n. As shown in this paper, this phenomenon occurs for strong orientations as well. In [4], the authors proved that, for every positive integer k, there exist innitely many graphs for which the metric dimension of any of its strongly-connected orientations is exactly k. They have also proved that there is no constant k such that the metric dimension of any tournament is at most k.

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Oriented trees in digraphs

Oriented trees in digraphs

k − 2, the digraph D[N + (u)] has at least (k − 2)(k − 4) + 1 > k−2 2  arcs, which is impossible since D is an oriented graph. Note that the analogue of Proposition 29 does not holds for digraphs (rather than oriented graphs). Indeed, there are connected digraphs such that d(v) ≥ 2k − 5 for every vertex v that do not contain every antidirected tree of order k of diameter 3. For example, let G be a regular bipartite graph of degree k − 3 with bipartition (A, B). Let D the digraph obtained from G by orienting all the edges from A to B and adding, for each a ∈ A (resp. b ∈ B) a copy of ~ K k−2 dominating a (resp. dominated by b). One can
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Oriented cliques and colorings of graphs with low maximum degree

Oriented cliques and colorings of graphs with low maximum degree

Abstract An oriented clique, or oclique, is an oriented graph G such that its oriented chromatic number χ o (G) equals its order |V (G)|. We disprove a conjecture of Duffy, MacGillivray, and Sopena [Oriented colourings of graphs with maximum degree three and four, Discrete Mathematics 342(4) (2019) 959–974] by showing that for maximum degree 4, the maximum order of an oclique is equal to 12. For maximum degree 5, we prove that the maximum order of an oclique is between 16 and 18. In the same paper, Duffy et al. also proved that the oriented chromatic number of connected oriented graphs with maximum degree 3 and 4 is at most 9 and 69, respectively. We improve these results by showing that the oriented chromatic number of non-necessarily connected oriented graphs with maximum degree 3 (resp. 4) is at most 9 (resp. 26). The bound of 26 actually follows from a general result which determines properties for a target graph to be universal for graphs of bounded maximum degree. This generalization also allows us to get the upper bound of 90 (resp. 306, 1322) for the oriented chromatic number of graphs with maximum degree 5 (resp. 6, 7).
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Packing Coloring of Undirected and Oriented Generalized Theta Graphs

Packing Coloring of Undirected and Oriented Generalized Theta Graphs

(mod 4). In this paper, we consider undirected graphs and orientations of undirected graphs, obtained by giving to each edge of such a graph one of its two possible orientations. The so-obtained oriented graphs are thus digraphs having no pair of opposite arcs. Let − → G be any orientation of an undirected graph G. Since for any two vertices u, v in V (G) we have d G (u, v) ≤ d − → G (u, v),

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Object-Oriented Database Benchmarks

Object-Oriented Database Benchmarks

type of systems, even if extensions are needed to take into account additional aspects, regard- ing Abstract Data Types, in particular. CONCLUSION The full specifications for the OCB object-oriented benchmark have been presented. OCB’s main qualities are its richness, its flexibility, and its compactness. This benchmark indeed offers an object base whose complexity has never been achieved before in object-oriented benchmarks. Furthermore, since this database and likewise the transactions running on it are wholly tunable through a collection of comprehensive but easily set parameters, OCB can be used to model many kinds of object-oriented database applications. Eventually, OCB’s code is short, reasonably easy to implement, and easily portable.
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Community-Oriented Technology Assessment

Community-Oriented Technology Assessment

Community-Oriented Technology Assessment The operational and economical sustainability of ICT are critical criteria for the impact analysis and technology projection activities of community-oriented technology assessment. Impact analyses must address the significant deficits in investment capital, infrastructure and experience that communities often face. The economics of a community will often preclude individualized solutions that are the norm in corporations, more affluent communities or developed nations as a whole. Instead, technology projections must include consideration of group-based solutions. Geographic factors may preclude certain modalities of communication in a community. Thus, novel approaches to using existing technologies will be necessary.
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Designing BRT-oriented development

Designing BRT-oriented development

Adopting an integrated approach to the design of transit routes and infrastructures, surrounding public spaces, real estate projects, public policy and governance, the students’ propos[r]

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Spy-Game on graphs

Spy-Game on graphs

Spy-Game on graphs 18 5 Appendix 5.1 With a countdown Fast robber on an infinite line: A robber walks at a speed of “2 hops per second” on Z. Around him, c guards are initially positionned “as needed”, and then walk at a speed of 1 hop per second. They must be able to catch him at a specific distant time t (i.e. be at distance O(1) from him). What is the asymptotic number of guards necessary to achieve that, when t grows large?

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On wheel-free graphs

On wheel-free graphs

configurations, see [15], that play a role in several theorems on the structure of graphs and matroids. Let us see more precisely how wheels play some role in the description of the structure of several graph classes. A hole in a graph is a chordless cycle on at least four vertices. The structure of a Berge graph G, that is a graph such that G and its complement do not contain odd holes, is studied in [3]. The results obtained there famously settled the Strong Perfect Graph Conjecture. The proof goes through several cases, and the last fifty pages of the proof deal with the case when G contains certain kinds of wheels. Consequently the structure of a Berge graph G is simpler when G does not contain these kinds of wheels. In addition, the structure of a graph G with no even holes is complex. A first decomposition theorem was given in [4] and a better one in [13]. In the later paper, very long arguments are devoted to situations when G contains certain kinds of wheels. This suggests that graphs that do not contain a wheel as an induced subgraph should have interesting structural properties. Understanding this structure might shed a new light on the works listed above. Since “understanding the structure” is a slightly fuzzy goal, we adress the following precise open questions.
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Contraction and deletion blockers for perfect graphs and H-free graphs

Contraction and deletion blockers for perfect graphs and H-free graphs

Theorem 6. For any fixed d ≥ 0, the d-Contraction Blocker(χ) problem can be solved in polynomial time for 3P 1 -free graphs. Proof. Let G be a graph with α(G) ≤ 2. Consider a colouring with χ(G) colours. The size of every colour class is at most 2. Hence every subgraph of G induced by two colour classes has at most 4 vertices, and as such has a spanning forest with in total at most 3 edges. This means that we can contract two colour classes to an independent set (that is, to a new colour class) by using at most three edge contractions. This observation gives us the following algorithm. We consider each set of at most three edge contractions, which we perform. Afterwards we decrease d by 1 and repeat this procedure until d = 0. For each resulting graph G  obtained in this way we check whether χ(G  ) ≤ χ(G) − d. If so, then the algorithm returns a yes-answer. Otherwise, that is, if χ(G  ) > χ(G) − d for each resulting graph G  , our algorithm returns a no-answer.
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Oriented trees in digraphs.

Oriented trees in digraphs.

In order to find an oriented tree T in digraphs of sufficiently large chromatic number, it would be useful to find a sequence of few removal of the in-leaves or out-leaves after which the tree is reduced to a single vertex. However, we do not know if finding such a sequence with the minimum number of steps can be done in polynomial time. Problem 10. What is the complexity of determining st(T ) for a given an oriented tree T ?

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On the hyperbolicity of bipartite graphs and intersection graphs

On the hyperbolicity of bipartite graphs and intersection graphs

5 Conclusion We have proved that the hyperbolicity of any bipartite graph can be approximated up to a small additive constant by only considering the smallest side of its bipartition. This means a decrease by half of the number of vertices to be considered, hence a speed-up in the computation of hyperbolicity. On a more theoretical side, we detailed a simple framework so as to bound the hyperbolicity of line graphs and several other intersection graphs. We let open the question whether our techniques could also be applied to more “exotic” generalizations of line graphs – say, edge clique graphs [10].
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Decomposing Berge graphs

Decomposing Berge graphs

This theorem is stronger than Theorems 1.2, 1.3 and 1.4 because path- cobipartite graphs may be seen either as graphs having a proper path 2-join (Theorems 1.2 and 1.3) or as a new basic class (Theorem 1.4). Path-double split graphs may be seen as graphs having a proper path 2-join (Theorems 1.2 and 1.3) or as graphs having a non-even skew partition (Theorem 1.4). And graphs having an homogeneous 2-join may be seen as graph having an homo- geneous pair (Theorems 1.4 and perhaps 1.2) or as graphs having a proper path 2-join (Theorems 1.3 and perhaps 1.2). Formally all these remarks are not always true: it may happen in special cases that path-cobipartite graphs and path-double split graphs have no proper 2-join. But such graphs are established in Lemma 2.3 to be basic or to have an even skew partition.
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Concept oriented biomedical information retrieval

Concept oriented biomedical information retrieval

Different from traditional ad hoc information retrieval task, medical IR is faced with an important challenge of vocabulary variation: although we have a set of well defined concepts, a [r]

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From Aspect-oriented Requirements Models to Aspect-oriented Business Process Design Models

From Aspect-oriented Requirements Models to Aspect-oriented Business Process Design Models

In [9], the authors propose Ram, an aspect-oriented mod- eling approach that provides multi-view modeling, covering structural, state-based, and scenario-based models. Adore and AoURN focus on the behavioral point of view based on scenarios and workflows with only limited structural mod- eling. However, the consistency of AoURN requirements and Adore design models can be greatly improved when they are combined with more detailed structural informa- tion as provided by Ram. A detailed structural view is also useful when extracting service interfaces from orchestration and fragment definitions and when applying composition al- gorithms such as Kompose [6] to obtain the service mod- els (e.g., with the help of Uml class diagrams). AoURN, on the other hand, offers intentional models that describe stakeholder goals as a complementary view of the reasons for system choices and decisions.
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