Class Of Digraphs Research Articles

Combinatorial abstraction refinement for feasibility analysis of static priorities

Combinatorial explosion is a challenge for many analysis problems in the theory of hard real-time systems. One of these problems is static priority schedulability of workload models which are more expressive than the traditional periodic task model. Different classes of directed graphs have been proposed in recent years to model structures like frames, branching and loops. In contrast to dynamic priority schedulers with pseudo-polynomial time analysis methods, static priority schedulability has been shown to be intractable since it is strongly coNP-hard already for the relatively simple class of cyclic digraphs. The core of this problem is the necessity to combine different behaviors of the participating tasks. We introduce a novel iterative approach to efficiently cope with this combinatorial explosion, called combinatorial abstraction refinement. In combination with other techniques it significantly reduces exponential growth of run-time for most inputs. We apply the method to static task priorities and demonstrate that a prototype implementation outperforms the state-of-the art pseudo-polynomial analysis for dynamic priority feasibility. It further shows better scaling behavior for randomly generated problem instances. We extend the approach to non-preemptive schedulers as well as static job-type priorities where jobs of different types in the same task may be assigned different static priorities. Finally, we provide a general, abstract formulation of the algorithm, since we believe that this method can be applicable to a variety of combinatorial problems in the theory of real-time systems with certain abstraction structures.

Distributed static output-feedback control for state synchronization in networks of identical LTI systems

This paper studies state synchronization of multi-agent systems with disturbances using distributed static output-feedback (OPFB) control. The bounded L2 gain synchronization problem using distributed static OPFB is defined and solved. The availability of only output measurements restricts the local controls design, while the communication graph topology restricts global information flow among the agents. It is shown here that these two types of restriction can be dealt with in a symmetric manner and lead to two similar conditions guaranteeing the existence of bounded L2 gain static OPFB. One condition is on the output measurement matrix on a local scale, and the other on the graph Laplacian matrix on a global scale. Under additional conditions a distributed two-player zero-sum game using static OPFB is also solved and leads to distributed Nash equilibrium on the communication graph. As a special case the static OPFB globally optimal control is given. A new class of digraphs satisfying the above condition on the graph Laplacian is studied. The synchronizing region for distributed static OPFB control is exposed and found to be conical, different than the infinite right-half plane synchronizing region for distributed state feedback.

The Upper Bound of Primitive Exponent of a Class of Special Nonnegative Matrix Pairs

There is a one-to-one relationship between nonnegative matrix pairs and two-colored digraph. With the knowledge of graph theory, by studying the associated directed digraph of a class of special nonnegative matrix pairs, that is a class of two-colored digraphs whose uncolored digraph have 3n-1 vertices and consists of one (3n-1)-cycle and one n-cycle are considered. The exponent and characteristic of extreme two-colored digraphs are given.

Directed elimination games

While tools from structural graph theory such as tree- or path-width have proved to be highly successful in coping with computational intractability on undirected graphs, corresponding width measures for directed graphs have not yet fulfilled their promise for broad algorithmic applications on directed graphs. One reason for this is that in most existing digraph width measures the class of acyclic digraphs has small width which immediately implies hardness of problems such as computing directed dominating sets.Fernau and Meister (2014) introduce the concept of elimination width and a corresponding graph searching game which overcomes this problem with acyclic digraphs. In their paper, the focus was on structural characterizations of classes of digraphs of bounded elimination width.In this paper we study elimination width from an algorithmic and graph searching perspective. We analyse variations of the elimination width game and show that this leads to width measures on which the directed dominating set problem and some variants of it become tractable.

An Isomorphism Theorem for Graphs

A seminal result by Lovasz states that two digraphs A and B (possibly with loops) are isomorphic if and only if for every digraph X the number of homomorphisms X → A equals the number of homomorphisms X → B. Lovasz used this result to deduce certain cancellation properties for the direct product of digraphs. We develop an analogous result for the class of digraphs without loops, and with weak homomorphisms replacing homomorphisms. We show that two digraphs A and B (without loops) are isomorphic if and only if the number of weak homomorphisms X → A equals the number of weak homomorphisms X → B. This result is then applied to deduce a general cancellation property for the strong product of digraphs as well as graphs.

A Graph Library for Isabelle

In contrast to other areas of mathematics such as calculus, number theory or probability theory, there is currently no standard library for graph theory for the Isabelle/HOL proof assistant. We present a formalization of directed graphs and essential related concepts. The library supports general infinite directed graphs (digraphs) with labeled and parallel arcs, but care has been taken not to complicate reasoning on more restricted classes of digraphs. We use this library to formalize a characterization of Euler Digraphs and to verify a method of checking Kuratowski subgraphs used in the LEDA library.

The Primitive Exponent of a Class of Special Nonnegative Matrix Pairs

There is a one-to-one relationship between nonnegative matrix pairs and two-colored digraph, so the problem of matrices can be changed into that of graphics to be solved. With the knowledge of graph theory, by studying the associated directed digraph of a class of special nonnegative matrix pairs, that is a class of two-colored digraphs whose uncolored digraph have vertices and consists of one-cycle and one-cycle are considered. The exponent and characteristic of extreme two-colored digraphs are given.

Vertex-disjoint directed and undirected cycles in general digraphs

The dicycle transversal numberτ(D) of a digraph D is the minimum size of a dicycle transversal of D, i.e. a set T⊆V(D) such that D−T is acyclic. We study the following problem: Given a digraph D, decide if there is a dicycle B in D and a cycle C in its underlying undirected graph UG(D) such that V(B)∩V(C)=∅. It is known that there is a polynomial time algorithm for this problem when restricted to strongly connected graphs, which actually finds B, C if they exist. We generalize this to any class of digraphs D with either τ(D)≠1 or τ(D)=1 and a bounded number of dicycle transversals, and show that the problem is NP-complete for a special class of digraphs D with τ(D)=1 and, hence, in general.

Cooperative Optimal Control for Multi-Agent Systems on Directed Graph Topologies

This note brings together stability and optimality theory to design distributed cooperative control protocols that guarantee consensus and are globally optimal with respect to a positive semi-definite quadratic performance criterion. A common problem in cooperative optimal control is that global optimization problems generally require global information, which is not available to distributed controllers. Optimal control for multi-agent systems is complicated by the fact that the communication graph topology interplays with the agent system dynamics. In the note we use an inverse optimality approach together with partial stability to consider the cooperative consensus and pinning control. Agents with identical linear time-invariant dynamics are considered. Communication graphs are assumed directed and having fixed topology. Structured quadratic performance indices are derived that capture the topology of the graph, which allows for global optimal control that is implemented using local distributed protocols. A new class of digraphs is defined that admits a distributed solution to the global optimal control problem, namely those with simple graph Laplacian matrices.

On the complexity of the 3-kernel problem in some classes of digraphs

Let D be a digraph with the vertex set V (D) and the arc set A(D). A subset N of V (D) is k-independent if for every pair of vertices u,v ∈ N, we have d(u,v),d(v,u) ≥ k; it is l-absorbent if for every u ∈ V (D) − N there exists v ∈ N such that d(u,v) ≤ l. A k-kernel of D is a k-independent and (k − 1)-absorbent subset ofV (D). A 2-kernel is called akernel. It is known that the problem of determining whether a digraph has a kernel (“the kernel problem”) is NP-complete, even in quiterestricted families of digraphs. In this paper we analyze the computational complexity of the corresponding 3-kernel problem, restricted to three natural families of digraphs. As a consequence of one of our main results we prove that the kernel problem remains NP-complete when restricted to 3-colorable digraphs.

Arc-Disjoint Paths in Decomposable Digraphs

We prove that the weak k-linkage problem is polynomial for every fixed k for totally i¾?-decomposable digraphs, under appropriate hypothesis on i¾?. We then apply this and recent results by Fradkin and Seymour on the weak k-linkage problem for digraphs of bounded independence number or bounded cut-width to get polynomial algorithms for some classes of digraphs like quasi-transitive digraphs, extended semicomplete digraphs, locally semicomplete digraphs all of which contain the class of semicomplete digraphs as a subclass and directed cographs.

On Panchromatic Digraphs and the Panchromatic Number

Let D = (V, A) be a simple finite digraph, and let $${\pi(D)}$$ ? ( D ) , the panchromatic number of D, be the maximum number of colours k such that for each (effective, or onto) colouring of its arcs $${\varsigma : A \to [k]}$$ ? : A ? [ k ] a monochromatic path kernel N ? V, as introduced in Galeana-Sanchez (Discrete Math 184: 87---99, 1998), exists. It is not hard to see that D has a kernel--in the sense of Von Neumann and Morgenstern (Theory of games and economic behaviour. Princeton University Press, Princeton, 1944)--if and only if $${\pi(D) = |A|}$$ ? ( D ) = | A | . In this note this invariant is introduced and some of its structural bounds are studied. For example, the celebrated theorem of Sands et al. (J Comb Theory Ser 33: 271---275, 1982), in terms of this invariant, settles that $${\pi(D) \geq 2}$$ ? ( D ) ? 2 . It will be proved that $$\pi(D) < |A| \iff \pi(D) < \min \left\{2\sqrt{\chi(D)}, \chi(L(D)), \theta(D) + \max {\text d}_c(K_i) + 1\right\},$$ ? ( D ) < | A | ? ? ( D ) < min 2 ? ( D ) , ? ( L ( D ) ) , ? ( D ) + max d c ( K i ) + 1 , where $${\chi(\cdot)}$$ ? ( · ) denotes the chromatic number, $${L(\cdot)}$$ L ( · ) denotes the line digraph, $${\theta(\cdot)}$$ ? ( · ) denotes the minimum partition into complete graphs of the underlying graph and $${\text{d}_c(\cdot)}$$ d c ( · ) denotes the dichromatic number. We also introduce the notion of a panchromatic digraph which is a digraph D such that for every k ≤ |A| and every k-colouring of its arcs, it has a monochromatic path kernel. Some classes of panchromatic digraphs are further characterised.

The complexity of multicut and mixed multicut problems in (di)graphs

The multicut problem with respect to the fixed pattern (di)graph Q is the following: given a (di)graph G, an integer k and a 1-1 mapping π of the vertices of Q to a subset X of V(G); determine whether there exists a set S of at most k edges (arcs) of G such that G−S has no (π(q),π(q′))-path for every edge (arc) qq′ of Q. We first prove a dichotomy result for the complexity (they are all either polynomial or NP-complete) for multicut problems in (di)graphs in terms of the pattern Q. Then we consider a variant where the pattern is a mixed graph M; the input is a digraph D, an integer k and a 1-1 mapping π from V(M) to some subset of V(D) and the goal is to decide whether there exists a set S of at most k arcs of D that meets every (π(q),π(q′))-path in D for every arc qq′ of M and every (π(p),π(p′))-path in the underlying undirected graph of D for every edge pp′ of M. Again we prove a dichotomy result for the complexity of such mixed multicut problems in terms of the pattern M. It turns out that the only polynomial cases are those where the pattern is either a polynomial graph pattern or a polynomial digraph pattern. As soon as M contains both edges and arcs, the M-mixed cut problem becomes NP-complete.Finally, we prove that the directed feedback arc set problem is NP-complete, even in the class of digraphs which have a feedback vertex set of size 2.

Parallel dynamical systems over special digraph classes

In a previous work, for parallel dynamical systems over digraphs corresponding to the simplest Boolean functions AND and OR, we proved that only fixed or eventually fixed points appear, as it occurs over undirected dependency graphs. Nevertheless, for general Boolean functions, it was shown that any period can appear, depending on the Boolean function that infers the global evolution operator of the system and on the structure of the dependency digraph. Motivated by these results, in this work, we analyse the orbit structure of parallel discrete dynamical systems over some special digraph classes.

The Exponent Set of a Class of Two-Colored Digraphs with One Common Vertex

A two-colored directed digraph D is primitive if and only if there exist nonnegative integers h and k with h+k&gt;0 such that for each pair (i,j) of vertices there is a (h,k)-walk in D from i to j. The exponent of the primitive two-colored digraph D is defined to be the smallest value of h+k over all suchand . With the knowledge of graph theory, a class of two-colored digraphs with two cycles whose uncolored digraph has 3n vertices and consists of one (2n+1)-cycle and one n-cycle is considered.The exponent bound, exponent set and characteristic of extral two-colored digraphs are given.

Signed Roman domination in digraphs

Let $$D$$D be a finite and simple digraph with vertex set $$V(D)$$V(D) and arc set $$A(D)$$A(D). A signed Roman dominating function (SRDF) on the digraph $$D$$D is a function $$f:V(D)\rightarrow \{-1,1,2\}$$f:V(D)?{-1,1,2} satisfying the conditions that (i) $$\sum _{x\in N^-[v]}f(x)\ge 1$$?x?N-[v]f(x)?1 for each $$v\in V(D)$$v?V(D), where $$N^-[v]$$N-[v] consists of $$v$$v and all in-neighbors of $$v$$v, and (ii) every vertex $$u$$u for which $$f(u)=-1$$f(u)=-1 has an in-neighbor $$v$$v for which $$f(v)=2$$f(v)=2. The weight of an SRDF $$f$$f is $$w(f)=\sum _{v\in V(D)}f(v)$$w(f)=?v?V(D)f(v). The signed Roman domination number $$\gamma _{sR}(D)$$?sR(D) of $$D$$D is the minimum weight of an SRDF on $$D$$D. In this paper we initiate the study of the signed Roman domination number of digraphs, and we present different bounds on $$\gamma _{sR}(D)$$?sR(D). In addition, we determine the signed Roman domination number of some classes of digraphs. Some of our results are extensions of well-known properties of the signed Roman domination number $$\gamma _{sR}(G)$$?sR(G) of graphs $$G$$G.

An upper bound for the energy of radial digraphs

Let D be a digraph with n vertices, a arcs, c2 closed walks of length 2 and spectral radius ρ(D). Recently Ayyaswamy, Balachandran and Gutman (2011) [1] proved that whenρ(D)⩾a+c22n⩾1 it is possible to construct an upper bound for the energy of digraphs, which improves the McClelland inequality for the energy of strongly connected digraphs given in Rada (2009) [16]. It is our interest in this paper to show that for general digraphs the inequalityρ(D)⩾a+c22n does not hold. However, we introduce the class of radial digraphs, that satisfy the spectral radius condition above, and for this class of digraphs we improve the bound for the energy given in Ayyaswamy et al. (2011) [1].

Formula omitted]-labeling of oriented planar graphs

The L(2,1)-labeling of a digraph D is a function l from the vertex set of D to the set of all nonnegative integers such that |l(x)−l(y)|≥2 if x and y are at distance 1, and l(x)≠l(y) if x and y are at distance 2, where the distance from vertex x to vertex y is the length of a shortest dipath from x to y. The minimum over all the L(2,1)-labelings of D of the maximum used label is denoted λ→(D). If C is a class of digraphs, the maximum λ→(D), over all D∈C is denoted λ→(C).In this paper we study the L(2,1)-labeling problem on oriented planar graphs providing some upper bounds on λ→. Then we focus on some specific subclasses of oriented planar graphs, improving the previous general bounds. Namely, for oriented prisms we compute the exact value of λ→, while for oriented Halin graphs and cacti we provide very close upper and lower bounds for λ→.

THE SIGNED STAR DOMINATION NUMBER OF CAYLEY GRAPHS

Let G be a simple connected graph with vertex set V(G) and edge set E(G). A function f : E(G) → {-1, 1} is called a signed star dominating function (SSDF) on G if ∑e∈E(v) f(e) ≥ 1 for every v ∈ V(G), where E(v) is the set of all edges incident to v. The signed star domination number of G is defined as γ SS (G) = min {∑e∈E(G) f(e) | f is a SSDF on G}. In this paper, we obtain exact values for the signed star domination number for certain classes of Cayley digraphs and Cayley graphs.

A Contribution to the Second Neighborhood Problem

Seymour's Second Neighborhood Conjecture asserts that every digraph (without digons) has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. It is proved for tournaments, tournaments missing a matching and tournaments missing a generalized star. We prove this conjecture for classes of digraphs whose missing graph is a comb, a complete graph minus 2 independent edges, or a complete graph minus the edges of a cycle of length 5.