Class Of Digraphs Research Articles

Signed double Roman domination numbers in digraphs

"Let $D=(V,A)$ be a finite simple digraph. A signed double Roman dominating function (SDRD-function) on the digraph $D$ is a function $f:V(D)\rightarrow\{-1,1,2, 3\}$ satisfying the following conditions: (i) $\sum_{x\in N^-[v]}f(x)\ge 1$ for each $v\in V(D)$, where $N^-[v]$ consist of $v$ and all in-neighbors of $v$, and (ii) if $f(v)=-1$, then the vertex $v$ must have at least two in-neighbors assigned 2 under $f$ or one in-neighbor assigned 3, while if $f(v)=1$, then the vertex $v$ must have at least one in-neighbor assigned 2 or 3. The weight of a SDRD-function $f$ is the value $\sum_{x\in V(D)}f(x)$. The signed double Roman domination number (SDRD-number) $\gamma_{sdR}(D)$ of a digraph $D$ is the minimum weight of a SDRD-function on $D$. In this paper we study the SDRD-number of digraphs, and we present lower and upper bounds for $\gamma_{sdR}(D)$ in terms of the order, maximum degree and chromatic number of a digraph. In addition, we determine the SDRD-number of some classes of digraphs."

Majority Colorings of Sparse Digraphs

A majority coloring of a directed graph is a vertex-coloring in which every vertex has the same color as at most half of its out-neighbors. Kreutzer, Oum, Seymour, van der Zypen and Wood proved that every digraph has a majority $4$-coloring and conjectured that every digraph admits a majority $3$-coloring. They observed that the Local Lemma implies the conjecture for digraphs of large enough minimum out-degree if, crucially, the maximum in-degree is bounded by a(n exponential) function of the minimum out-degree.
 Our goal in this paper is to develop alternative methods that allow the verification of the conjecture for natural, broad digraph classes, without any restriction on the in-degrees. Among others, we prove the conjecture 1) for digraphs with chromatic number at most $6$ or dichromatic number at most $3$, and thus for all planar digraphs; and 2) for digraphs with maximum out-degree at most $4$. The benchmark case of $r$-regular digraphs remains open for $r \in [5,143]$.
 Our inductive proofs depend on loaded inductive statements about precoloring extensions of list-colorings. This approach also gives rise to stronger conclusions, involving the choosability version of majority coloring.
 We also give further evidence towards the existence of majority-$3$-colorings by showing that every digraph has a fractional majority 3.9602-coloring. Moreover we show that every digraph with large enough minimum out-degree has a fractional majority $(2+\varepsilon)$-coloring.

Digraphs with proper connection number two

A directed path in a digraph is proper if any two consecutive arcs on the path have distinct colors. An arc-colored digraph D is proper connected if for any two distinct vertices x and y of D, there are both proper (x,y)-directed paths and proper (y,x)-directed paths in D. The proper connection number pc→(D) of a digraph D is the minimum number of colors that can be used to make D proper connected. Obviously, if a digraph has a proper connection number, it must be strongly connected, and pc→(D)=1 if and only if D is complete. Magnant et al. showed that pc→(D)≤3 for all strong digraphs D, and Ducoffe et al. proved that deciding whether a given digraph has proper connection number at most two is NP-complete. In this paper, we give a few classes of strong digraphs with proper connection number two, and from our proofs one can construct an optimal arc-coloring for a digraph of order n in time O(n3).

Chordality of locally semicomplete and weakly quasi-transitive digraphs

Chordal graphs are important in the structural and algorithmic graph theory. A digraph analogue of chordal graphs was introduced by Haskin and Rose in 1973 but has not been the subject of active studies until recently when a characterization of semicomplete chordal digraphs in terms of forbidden subdigraphs was found by Meister and Telle.Locally semicomplete digraphs, quasi-transitive digraphs, and extended semicomplete digraphs are amongst the most popular generalizations of semicomplete digraphs. We extend the forbidden subdigraph characterization of semicomplete chordal digraphs to locally semicomplete chordal digraphs. We introduce a new class of digraphs, called weakly quasi-transitive digraphs, which contains quasi-transitive digraphs, symmetric digraphs, and extended semicomplete digraphs, but is incomparable to the class of locally semicomplete digraphs. We show that weakly quasi-transitive digraphs can be recursively constructed by simple substitutions from transitive oriented graphs, semicomplete digraphs, and symmetric digraphs. This recursive construction of weakly quasi-transitive digraphs, similar to the one for quasi-transitive digraphs, demonstrates the naturalness of the new digraph class. As a by-product, we prove that the forbidden subdigraphs for semicomplete chordal digraphs are the same for weakly quasi-transitive chordal digraphs. The forbidden subdigraph characterization of weakly quasi-transitive chordal digraphs generalizes not only the recent results on quasi-transitive chordal digraphs and extended semicomplete chordal digraphs but also the classical results on chordal graphs.

On the restricted numerical range of the Laplacian matrix for digraphs

ABSTRACT In this article, we present the restricted numerical for the Laplacian matrix of a directed graph (digraph). We motivate our interest in the restricted numerical range by its close connection to the algebraic connectivity of a digraph. Moreover, we show that the restricted numerical range can be used to characterize digraphs, some of which are not determined by their Laplacian spectrum. Finally, we identify a new class of digraphs that are characterized by having a real restricted numerical range.

Min-Orderable Digraphs

We unify several seemingly different graph and digraph classes under one umbrella. These classes are all, broadly speaking, different generalizations of interval graphs, and include, in addition to interval graphs, adjusted interval digraphs, complements of threshold tolerance graphs (known as co-TT graphs), bipartite interval containment graphs, bipartite co-circular arc graphs, and two-directional orthogonal ray bigraphs. (The last three classes coincide, but have been investigated in different contexts.) We show that all of the above classes are united by a common ordering characterization, the existence of a min ordering. However, because the presence or absence of reflexive relationships (loops) affects whether a graph or digraph has a min ordering, to obtain this result, we must define the graphs and digraphs to have those loops that are implied by their definitions. These have been largely ignored in previous work. We propose a common generalization of all these graph and digraph classes, namely signed-interval digraphs, characterized by the existence of a compact representation, a signed-interval model, which is a generalization of known representations of the graph classes. We show that the signed-interval digraphs are precisely those digraphs that are characterized by the existence of a min ordering when the loops implied by the model are considered part of the graph. We also offer an alternative geometric characterization of these digraphs. We show that co-TT graphs are the symmetric signed-interval digraphs, the adjusted interval digraphs are the reflexive signed-interval digraphs, and the interval graphs are the intersection of these two classes, namely, the reflexive and symmetric signed-interval digraphs. Similar results hold for bipartite interval containment graphs, bipartite co-circular arc graphs, and two-directional orthogonal ray bigraphs.

Mod difference labeling of some classes of digraphs

Mod difference labeling of some classes of digraphs

Total Roman domination in digraphs

Let D be a finite and simple digraph with vertex set V (D). A Roman dominating function (RDF) on a digraph D is a function f : V (D) → {0, 1, 2} satisfying the condition that every vertex v with f (v) = 0 has an in-neighbor u with f (u) = 2. The weight of an RDF f is the value . An RDF f on D with no isolated vertex is called a total Roman dominating function if the subdigraph of D induced by the set {v ∈ V (D): f (v) ≠ 0} has no isolated vertex. The Roman domination number is the minimum weight of a total Roman dominating function on D. In this paper, we initiate the study of the total Roman domination number in digraphs and show its relationship to other domination parameters. In particular, we present some sharp bounds for the total Roman domination number and we determine the total Roman domination number of some classes of digraphs.

Almost Complete and Equable Heteroclinic Networks

Heteroclinic connections are trajectories that link invariant sets for an autonomous dynamical flow: these connections can robustly form networks between equilibria, for systems with flow-invariant spaces. In this paper we examine the relation between the heteroclinic network as a flow-invariant set and directed graphs of possible connections between nodes. We consider realizations of a large class of transitive digraphs as robust heteroclinic networks and show that although robust realizations are typically not complete (i.e. not all unstable manifolds of nodes are part of the network), they can be almost complete (i.e. complete up to a set of zero measure within the unstable manifold) and equable (i.e. all sets of connections from a node have the same dimension). We show there are almost complete and equable realizations that can be closed by adding a number of extra nodes and connections. We discuss some examples and describe a sense in which an equable almost complete network embedding is an optimal description of stochastically perturbed motion on the network.

Extremal iota energy of a subclass of bicyclic digraphs and sidigraphs

The iota energy of an [Formula: see text]-vertex digraph [Formula: see text] is defined by [Formula: see text], where [Formula: see text] are eigenvalues of [Formula: see text] and [Formula: see text] is the imaginary part of eigenvalue [Formula: see text] Recently, this concept of iota energy of digraphs is extended to sidigraphs. The iota energy of a sidigraph is defined in an analogous way. In this paper, we consider a class of digraphs with two linear subdigraphs of equal length and find digraphs in this class with extremal iota energy. Furthermore, we take a similar class of sidigraphs and find minimal and maximal iota energy among the sidigraphs in this class.

Formula omitted]-panchromatic digraphs

Let H and D be two digraphs; D without loops or multiple arcs. An H−coloring of D is a function ρ:A(D)→V(H). We say that D is an (H,ρ)−colored digraph. For an arc (x,y) of D, we say that ρ(x,y) is the color of (x,y) over the H−coloring ρ.A directed path (x1,…,xn) in D is an (H,ρ)−path if (ρ(x1,x2),…,ρ(xn−1,xn)) is a directed walk in H. An (H,ρ)−kernel in an (H,ρ)−colored digraph is a subset of vertices of D, say S, such that for every pair of different vertices in S there is no (H,ρ)−path between them and every vertex outside S can reach S by an (H,ρ)−path. A digraph D is an ℋ-panchromatic digraph if D has an (H,ρ)−kernel for every digraph H and every H−coloring ρ of D. In this paper we show that ℋ-panchromatic digraphs cannot be characterized by means of certain forbidden subdigraphs. Also we will show ℋ-panchromaticity of some classes of digraphs and we show that ℋ-panchromaticity can be hereditary in some operations of digraphs.

Characterization of color patterns by dynamic [formula omitted]-paths

Let H be a digraph possibly with loops, let D be a multidigraph, and let c:A(D)→V(H) be a coloring of the arcs of D with the vertices of H. A sequence W of vertices in D, W=(x0,x1,…,xn), is a dynamic H-walk if for each i∈{0,…,n−2} there exist fi and fi+1 such that fi is an arc from xi to xi+1, fi+1 is an arc from xi+1 to xi+2, and (c(fi),c(fi+1)) is an arc of H. If W does not repeat vertices, W is a dynamic H-path. For u,v∈V(D), we say that u reaches v by dynamic H-paths if there exists a dynamic H-path from u to v in D. A subset K⊆V(D) is a kernel by dynamic H-paths of D if every vertex in V(D)−K reaches some vertex in K by dynamic H-paths (absorbent by dynamic H-paths), and no vertex in K can be reached by another vertex in K by dynamic H-paths in D (independent by dynamic H-paths).Let D̃3 be the family of digraphs H such that for every H-arc-colored multidigraph D, D has a kernel by dynamic H-paths. Let D̃2 (resp. D̃1) be the class of digraphs H such that for every H-arc-colored multidigraph D (resp. multitournament), D has an independent absorbent by dynamic H-paths set. In this work, we provide a characterization of D̃2 and show that D̃3 is one of two families.

Extremal Digraphs for an Upper Bound on the Double Roman Domination Number

A doubleRoman dominating function of a digraph D is a function $$f:V\longrightarrow \{0,1,2,3\}$$ such that every vertex u for which $$f(u)=0$$ has an in-neighbor v for which $$f(v)=3$$ or at least two in-neighbors assigned 2 under f, while if $$f(u)=1$$, then the vertex u must have at least one in-neighbor assigned 2 or 3. The weight of a double Roman dominating function is the value $$f(V)=\sum _{u\in V}f(u)$$. The minimum weight of a double Roman dominating function of a digraph D is called the double Roman domination number of D, denoted by $$\gamma _{dR}\left( D\right) $$. This paper gives a descriptive characterization for some classes of digraphs satisfying $$\gamma _{dR}(D)=2\left( n-\Delta ^{+}(D)\right) +1$$. Also, a descriptive characterization for digraphs D of order $$n\ge 4$$ for which $$\gamma _{dR}(D)+\gamma _{dR}({\overline{D}})=2n+3$$ holds where $${\overline{D}}$$ is the complement of D.

The star dichromatic number

We introduce a new notion of circular colourings for digraphs. The idea of this quantity, called star dichromatic number $\vec{\chi}^\ast(D)$ of a digraph $D$, is to allow a finer subdivision of digraphs with the same dichromatic number into such which are easier or harder to colour by allowing fractional values. This is related to a coherent notion for the vertex arboricity of graphs introduced by Wang et al. and resembles the concept of the star chromatic number of graphs introduced by Vince in the framework of digraph colouring. After presenting basic properties of the new quantity, including range, simple classes of digraphs, general inequalities and its relation to integer counterparts as well as other concepts of fractional colouring, we compare our notion with the notion of circular colourings for digraphs introduced by Bokal et al. and point out similarities as well as differences in certain situations. As it turns out, the star dichromatic number is a lower bound for the circular dichromatic number of Bokal et al., but the gap between the numbers may be arbitrarily close to $1$. We conclude with a discussion of the case of planar digraphs and point out some open problems.

Triangle edge deletion on planar glasses-free RGB-digraphs

In this paper, we revisit the triangle edge deletion problem a variant of feedback arc set problem which is one of the Karp's 21 NP-complete problems. Given a graph G(V,E), it is to find the edge set F such that |F|≤k and G′(V,E∖F) is a triangle-free graph. Yannakakis [1] showed it is hard for simple digraphs as well. The latest results showed that triangle edge deletion problem is still hard even for (a) planar undirected graphs of maximum degree seven [2] and (b) RGB-digraphs [3] which is a special class of digraphs. To make a further step, we show a new result in this paper that triangle edge deletion problem is still NP-complete even for planar RGB-digraphs of maximum degree eight with no glasses. On the other hand, we can find a kernel consisting of 11k/3 vertices for planar RGB-digraphs with no glasses. The results have an interesting application for the derivation of a tighter dichotomy on resilience decision problem in database theory.

On average distance in tournaments and Eulerian digraphs

We present upper bounds on the average distance for two important classes of digraphs, viz., tournaments and Eulerian digraphs. We first show that the average distance of Eulerian digraphs of order n and minimum degree δ is bounded from above by 32(δ+1)n+32+1n−1. The coefficient 32(δ+1) is close to being optimal. We also give an improved bound for Eulerian bipartite digraphs. We then give upper bounds on the average distance of tournaments in terms of order and edge-connectivity, and in terms of diameter only. Both bounds are sharp apart from an additive constant.

Strong immersion is a well‐quasi‐ordering for semicomplete digraphs

AbstractWe prove that the strong immersion order is a well‐quasi‐ordering on the class of semicomplete digraphs, thereby strengthening a result of Chudnovsky and Seymour (2011, J. Comb. Theory, Series B, 101, 47–53) that this holds for the class of tournaments.

Degree constrained 2-partitions of semicomplete digraphs

A 2-partition of a digraph D is a partition (V1,V2) of V(D) into two disjoint non-empty sets V1 and V2 such that V1∪V2=V(D). A semicomplete digraph is a digraph with no pair of non-adjacent vertices. We consider the complexity of deciding whether a given semicomplete digraph has a 2-partition such that each part of the partition induces a (semicomplete) digraph with some specified property. In [4] and [5] Bang-Jensen, Cohen and Havet determined the complexity of 120 such 2-partition problems for general digraphs. Several of these problems are NP-complete for general digraphs and thus it is natural to ask whether this is still the case for well-structured classes of digraphs, such as semicomplete digraphs. This is the main topic of the paper. More specifically, we consider 2-partition problems where the set of properties are minimum out-, minimum in- or minimum semi-degree. Among other results we prove the following:•For all integers k1,k2≥1 and k1+k2≥3 it is NP-complete to decide whether a given digraph D has a 2-partition (V1,V2) such that D〈Vi〉 has out-degree at least ki for i=1,2.•For every fixed choice of integers α,k1,k2≥1 there exists a polynomial algorithm for deciding whether a given digraph of independence number at most α has a 2-partition (V1,V2) such that D〈Vi〉 has out-degree at least ki for i=1,2.•For every fixed integer k≥1 there exists a polynomial algorithm for deciding whether a given semicomplete digraph has a 2-partition (V1,V2) such that D〈V1〉 has out-degree at least one and D〈V2〉 has in-degree at least k.•It is NP-complete to decide whether a given semicomplete digraph D has a 2-partition (V1,V2) such that D〈Vi〉 is a strong tournament.

Chronological rectangle digraphs which are two-terminal series–parallel

Chronological interval digraphs are a natural directed analogue of interval graphs. They admit elegant characterizations as well as efficient recognition algorithms. Recently, a generalization of chronological interval digraphs to 2-dimensional space called chronological rectangle digraphs was introduced and studied. Although many interesting properties of chronological rectangle digraphs have been discovered, the most fundamental question of whether this class of digraphs can be recognized in polynomial-time remains unanswered. This is largely due to the lack of a good structural characterization for the class. In this paper, we study chronological rectangle digraphs which are also two-terminal series–parallel. We show that this restricted class of digraphs admits a forbidden induced subdigraph characterization, which leads to a polynomial-time recognition algorithm for this class of digraphs.

Twin signed total Roman domination numbers in digraphs

Let [Formula: see text] be a finite simple digraph with vertex set [Formula: see text] and arc set [Formula: see text]. A twin signed total Roman dominating function (TSTRDF) on the digraph [Formula: see text] is a function [Formula: see text] satisfying the conditions that (i) [Formula: see text] and [Formula: see text] for each [Formula: see text], where [Formula: see text] (respectively [Formula: see text]) consists of all in-neighbors (respectively out-neighbors) of [Formula: see text], and (ii) every vertex [Formula: see text] for which [Formula: see text] has an in-neighbor [Formula: see text] and an out-neighbor [Formula: see text] with [Formula: see text]. The weight of an TSTRDF [Formula: see text] is [Formula: see text]. The twin signed total Roman domination number [Formula: see text] of [Formula: see text] is the minimum weight of an TSTRDF on [Formula: see text]. In this paper, we initiate the study of twin signed total Roman domination in digraphs and we present some sharp bounds on [Formula: see text]. In addition, we determine the twin signed Roman domination number of some classes of digraphs.