Gregory Gutin

• PhD
• Professor (Full) at Royal Holloway University of London

For any arc-weighted oriented graph $D=(V(D), A(D),w)$, we write ${\rm fas}_w(D)$ to denote the minimum weight of a feedback arc set in $D$. In this paper, we consider upper bounds on ${\rm fas}_w(D)$ for arc-weight oriented graphs $D$ with bounded maximum degrees and directed girth. We obtain such bounds by introducing a new parameter ${\rm fasd}(...

Erd{\H o}s (1963) initiated extensive graph discrepancy research on 2-edge-colored graphs. Gishboliner, Krivelevich, and Michaeli (2023) launched similar research on oriented graphs. They conjectured the following generalization of Dirac's theorem: If the minimum degree $\delta$ of an $n$-vertex oriented graph $G$ is greater or equal to $n/2$,then...

Role mining is a technique that is used to derive a role-based authorization policy from an existing policy. Given a set of users U , a set of permissions P and a user-permission authorization relation UPA ⊆ U × P , a role mining algorithm seeks to compute a set of roles R , a user-role authorization relation UA ⊆ U × R and a permission-role author...

An oriented multigraph is a directed multigraph without directed 2-cycles. Let ${\rm fas}(D)$ denote the minimum size of a feedback arc set in an oriented multigraph $D$. The degree of a vertex is the sum of its out- and in-degrees. In several papers, upper bounds for ${\rm fas}(D)$ were obtained for oriented multigraphs $D$ with maximum degree upp...

A run of the deferred acceptance (DA) algorithm may contain proposals that are sure to be rejected. We introduce the accelerated deferred acceptance algorithm that proceeds in a similar manner to DA but with sure-to-be rejected proposals ruled out. Accelerated deferred acceptance outputs the same stable matching as DA but does so more efficiently:...

An oriented graph $D$ is converse invariant if, for any tournament $T$, the number of copies of $D$ in $T$ is equal to that of its converse $-D$. El Sahili and Ghazo Hanna [J. Graph Theory 102 (2023), 684-701] showed that any oriented graph $D$ with maximum degree at most 2 is converse invariant. They proposed a question: Can we characterize all co...

A bisection in a graph is a cut in which the number of vertices in the two parts differ by at most 1. In this paper, we give lower bounds for the maximum weight of bisections of edge-weighted graphs with bounded maximum degree. Our results improve a bound of Lee, Loh, and Sudakov (J. Comb. Th. Ser. B 103 (2013)) for (unweighted) maximum bisections...

In 1981, Bermond and Thomassen conjectured that for any positive integer $k$, every digraph with minimum out-degree at least $2k-1$ admits $k$ vertex-disjoint directed cycles. In this short paper, we verify the Bermond-Thomassen conjecture for triangle-free multipartite tournaments and 3-partite tournaments. Furthermore, we characterize 3-partite t...

For a vertex of a digraph, (, respectively) is the number of vertices at distance 1 from (to, respectively) and is the number of vertices at distance 2 from . In 1995, Seymour conjectured that for any oriented graph there exists a vertex such that . In 2006, Sullivan conjectured that there exists a vertex in such that . We give a sufficient conditi...

We investigate the difficulty of finding economically efficient solutions to coordination problems on graphs. Our work focuses on two forms of coordination problem: pure-coordination games and anti-coordination games. We consider three objectives in the context of simple binary-action polymatrix games: (i) maximizing welfare, (ii) maximizing potent...

We introduce and study a new optimization problem on digraphs, termed Maximum Weighted Digraph Partition (MWDP) problem. We prove three complexity dichotomies for MWDP: on arbitrary digraphs, on oriented digraphs, and on symmetric digraphs. We demonstrate applications of the dichotomies for binary-action polymatrix games and several graph theory pr...

For a vertex $x$ of a digraph, $d^+(x)$ ($d^-(x)$, resp.) is the number of vertices at distance 1 from (to, resp.) $x$ and $d^{++}(x)$ is the number of vertices at distance 2 from $x$. In 1995, Seymour conjectured that for any oriented graph $D$ there exists a vertex $x$ such that $d^+(x)\leq d^{++}(x)$. In 2006, Sullivan conjectured that there exi...

We investigate the difficulty of finding economically efficient solutions to coordination problems on graphs. Our work focuses on two forms of coordination problem: pure-coordination games and anti-coordination games. We consider three objectives in the context of simple binary-action polymatrix games: (i) maximizing welfare, (ii) maximizing potent...

Witnessing the advancing scale and complexity of chip design and benefiting from high-performance computation technologies, the simulation of Very Large Scale Integration (VLSI) Circuits imposes an increasing requirement for acceleration through parallel computing with GPU devices. However, the conventional parallel strategies do not fully align wi...

We obtain lower and upper bounds for the maximum weight of a directed cut in the classes of weighted digraphs and weighted acyclic digraphs as well as in some of their subclasses. We compare our results with those obtained for the maximum size of a directed cut in unweighted digraphs. In particular, we show that a lower bound obtained by Alon, Boll...

Cartesian product networks are always regarded as a tool for ``combining'' two given networks with established properties to obtain a new one that inherits properties from both. For a graph $F=(V,E)$ and a set $S\subseteq V(F)$ of at least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner tree connecting $S$} (or simply, \emph{an $S$-tree...

An orientation D of G is proper if for every \(xy\in E(G)\), we have \(d^-_D(x)\ne d^-_D(y)\). An orientation D is a p-orientation if the maximum in-degree of a vertex in D is at most p. The minimum integer p such that G has a proper p-orientation is called the proper orientation number pon(G) of G [introduced by Ahadi and Dehghan (Inf Process Lett...

We introduce and study two natural generalizations of the Connected Vertex Cover (VC) problem: the p-Edge-Connected and p-Vertex-Connected VC problem (where p≥2 is a fixed integer). We obtain an 2O(pk)nO(1)-time algorithm for p-Edge-Connected VC and an 2O(k2)nO(1)-time algorithm for p-Vertex-Connected VC. Thus, like Connected VC, both constrained V...

Let D be a digraph and let λ(D) denote the number of vertices in a longest path of D. For a pair of vertex-disjoint induced subdigraphs A and B of D, we say that (A,B) is a partition of D if V(A)∪V(B)=V(D). The Path Partition Conjecture (PPC) states that for every digraph, D, and every integer q with 1≤q≤λ(D)−1, there exists a partition (A,B) of D...

In this paper, we study two types of strong subgraph packing problems in digraphs, including internally disjoint strong subgraph packing problem and arc-disjoint strong subgraph packing problem. These problems can be viewed as generalizations of the famous Steiner tree packing problem and are closely related to the strong arc decomposition problem....

An instance I of the Stable Matching Problem (SMP) is given by a bipartite graph with a preference list of neighbors for every vertex. A swap in I is the exchange of two consecutive vertices in a preference list. A swap can be viewed as a smallest perturbation of I. Boehmer et al. (2021) designed a polynomial-time algorithm to find the minimum numb...

The traveling salesman problem (TSP) is one of the classic research topics in the field of operations research, graph theory and computer science. In this paper, we propose a generalized model of traveling salesman problem, denoted by generalized traveling salesman path problem. Let G=(V,E,c) be a weighted complete graph, in which c is a nonnegativ...

A graph $H$ is a clique graph if $H$ is a vertex-disjoin union of cliques. Abu-Khzam (2017) introduced the $(a,d)$-{Cluster Editing} problem, where for fixed natural numbers $a,d$, given a graph $G$ and vertex-weights $a^*:\ V(G)\rightarrow \{0,1,\dots, a\}$ and $d^*{}:\ V(G)\rightarrow \{0,1,\dots, d\}$, we are to decide whether $G$ can be turned...

A directed graph D is semicomplete if for every pair x, y of vertices of D, there is at least one arc between x and y. Thus, a tournament is a semicomplete digraph. In the Directed Component Order Connectivity (DCOC) problem, given a digraph \(D=(V,A)\) and a pair of natural numbers k and \(\ell \), we are to decide whether there is a subset X of V...

Branchings play an important role in digraph theory and algorithms. In particular, a chapter in the monograph of Bang-Jensen and Gutin, Digraphs: Theory, Algorithms and Application, Ed. 2, 2009 is wholly devoted to branchings. The well-known Edmonds Branching Theorem provides a characterization for the existence of k arc-disjoint out-branchings roo...

We prove the following new results. (a)Let T be a regular tournament of order 2n+1≥11 and S a subset of V(T). Suppose that |S|≤12(n−2) and x, y are distinct vertices in V(T)∖S. If the subtournament T−S contains an (x,y)-path of length r, where 3≤r≤|V(T)∖S|−2, then T−S also contains an (x,y)-path of length r+1. (b)Let T be an m-irregular tournament...

A {\em quasi-kernel} of a digraph $D$ is an independent set $Q\subseteq V(D)$ such that for every vertex $v\in V(D)\backslash Q$, there exists a directed path with one or two arcs from $v$ to a vertex $u\in Q$. In 1974, Chv\'{a}tal and Lov\'{a}sz proved that every digraph has a quasi-kernel. In 1976, Erd\H{o}s and S\'zekely conjectured that every s...

We survey the field of algorithms and complexity for graph problems parameterized above or below guaranteed values, a research area which was pioneered by Venkatesh Raman. Those problems seek, for a given graph $G$, a solution whose value is at least $g(G)+k$ or at most $g(G)-k$, where $g(G)$ is a guarantee on the value that any solution on $G$ tak...

As an algorithmic framework, message passing is extremely powerful and has wide applications in the context of different disciplines including communications, coding theory, statistics, signal processing, artificial intelligence and combinatorial optimization. In this paper, we investigate the performance of a message-passing algorithm called min-s...

Let G $G$ be a graph on n $n$ vertices. For i∈{0,1} $i\in \{0,1\}$, a spanning forest F $F$ of G $G$ is called an i $i$‐perfect forest if every tree in F $F$ is an induced subgraph of G $G$ and exactly i $i$ vertices of F $F$ have even degree (including zero). An i $i$‐perfect forest of G $G$ is proper if it has no vertices of degree zero. Scott sh...

Recent work has shown that many problems of satisfiability and resiliency in workflows may be viewed as special cases of the authorization policy existence problem (APEP), which returns an authorization policy if one exists and “No” otherwise. However, in many practical settings it would be more useful to obtain a “least bad” policy than just a “No...

In this paper we consider stable matchings that are subject to assignment constraints. These are matchings that require certain assigned pairs to be included, insist that some other assigned pairs are not, and, importantly, are stable. Our main contribution is an algorithm that determines when assignment constraints are compatible with stability. W...

We introduce a novel approach of using important cuts which allowed us to design significantly faster fixed-parameter tractable (FPT) algorithms for the following routing problems: the Mixed Chinese Postman Problem parameterized by the number of directed edges (Gutin et al., JCSS 2017), the Minimum Shared Edges problem (MSE) parameterized by the nu...

Let $D=(V,A)$ be a digraph of order $n$, $S$ a subset of $V$ of size $k$ and $2\le k\leq n$. A strong subgraph $H$ of $D$ is called an $S$-strong subgraph if $S\subseteq V(H)$. A pair of $S$-strong subgraphs $D_1$ and $D_2$ are said to be arc-disjoint if $A(D_1)\cap A(D_2)=\emptyset$. Let $\lambda_S(D)$ be the maximum number of arc-disjoint $S$-str...

In this paper, we investigate the performance of message-passing algorithms for the weighted min–max flow (WMMF) problem which was introduced by Ichimori et al. (1980). WMMF was well studied in combinational optimization, as it provides important applications in time transportation problem and the storage management problem. We develop a message-pa...

The workflow satisfiability problem (WSP) is a well-studied problem in access control seeking allocation of authorised users to every step of the workflow, subject to workflow specification constraints. It was noticed that the number $k$ of steps is typically small compared to the number of users in the real-world instances of WSP; therefore $k$...

An instance $I$ of the Stable Matching Problem (SMP) is given by a bipartite graph with a preference list of neighbors for every vertex. A swap in $I$ is the exchange of two consecutive vertices in a preference list. A swap can be viewed as a smallest perturbation of $I$. Boehmer et al. (2021) designed a polynomial-time algorithm to find the minimu...

We prove the following new results. (a) Let $T$ be a regular tournament of order $2n+1\geq 11$ and $S$ a subset of $V(T)$. Suppose that $|S|\leq \frac{1}{2}(n-2)$ and $x$, $y$ are distinct vertices in $V(T)\setminus S$. If the subtournament $T-S$ contains an $(x,y)$-path of length $r$, where $3\leq r\leq |V(T)\setminus S|-2$, then $T-S$ also contai...

Branchings play an important role in digraph theory and algorithms. In particular, a chapter in the monograph of Bang-Jensen and Gutin, Digraphs: Theory, Algorithms and Application, Ed. 2, 2009 is wholly devoted to branchings. The well-known Edmonds Branching Theorem provides a characterization for the existence of k arc-disjoint out-branchings roo...

Hamiltonian path problem is one of the fundamental problems in graph theory, the aim is to find a path in the graph that visits each vertex exactly once. In this paper, we consider a generalizedized problem: given a complete weighted undirected graph \(G=(V,E,c)\), two specified vertices s and t, let \(V^{\prime }\) and \(E^{\prime }\) be subsets o...

Let $D$ be a digraph and let $\lambda(D)$ denote the number of vertices in a longest path of $D$. For a pair of vertex-disjoint induced subdigraphs $A$ and $B$ of $D$, we say that $(A,B)$ is a partition of $D$ if $V(A)\cup V(B)=V(D).$ The Path Partition Conjecture (PPC) states that for every digraph, $D$, and every integer $q$ with $1\leq q\leq\lam...

In this paper, we study two types of strong subgraph packing problems in digraphs, including internally disjoint strong subgraph packing problem and arc-disjoint strong subgraph packing problem. These problems can be viewed as generalizations of the famous Steiner tree packing problem and are closely related to the strong arc decomposition problem....

User authorization queries in the context of role-based access control have attracted considerable interest in the past 15 years. Such queries are used to determine whether it is possible to allocate a set of roles to a user that enables the user to complete a task, in the sense that all the permissions required to complete the task are assigned to...

In his paper “Kings in Bipartite Hypertournaments,” Petrovic stated two conjectures on 4-kings in multipartite hypertournaments. We prove one of these conjectures and give counterexamples for the other.

How does social distancing affect the reach of an epidemic in social networks? We present Monte Carlo simulation results of a susceptible–infected–removed with social distancing model. The key feature of the model is that individuals are limited in the number of acquaintances that they can interact with, thereby constraining disease transmission to...

Due to their ubiquity and extensive applications, graph routing problems have been widely studied in the fields of operations research, computer science and engineering. An important example of a routing problem is the traveling salesman problem. In this paper, we consider two variants of the general cluster routing problem. In these variants, we a...

We provide necessary and sufficient conditions on the preferences of market participants for a unique stable matching in models of two-sided matching with non-transferable utility. We use the process of iterated deletion of unattractive alternatives (IDUA), a formalisation of the reduction procedure in Balinski and Ratier (1997), and we show that a...

In this survey we overview known results on the strong subgraph [Formula: see text]-connectivity and strong subgraph [Formula: see text]-arc-connectivity of digraphs. After an introductory section, the paper is divided into four sections: basic results, algorithms and complexity, sharp bounds for strong subgraph [Formula: see text]-(arc-)connectivi...

Problems of satisfiability and resiliency in workflows have been widely studied in the last decade. Recent work has shown that many such problems may be viewed as special cases of the authorization policy existence problem (APEP), which returns an authorization policy if one exists and 'No' otherwise. A solution may not exist because of the restric...

The workflow satisfiability problem (WSP) is a well-studied problem in access control seeking allocation of authorised users to every step of the workflow, subject to workflow specification constraints. It was noticed that the number $k$ of steps is typically small compared to the number of users in the real-world instances of WSP; therefore $k$ is...

Let $$D=(V,A)$$ D = ( V , A ) be a digraph of order n , S a subset of V of size k and $$2\le k\le n$$ 2 ≤ k ≤ n . A strong subgraph H of D is called an S - strong subgraph if $$S\subseteq V(H)$$ S ⊆ V ( H ) . A pair of S -strong subgraphs $$D_1$$ D 1 and $$D_2$$ D 2 are said to be arc-disjoint if $$A(D_1)\cap A(D_2)=\emptyset$$ A ( D 1 ) ∩ A ( D 2...

Let $G$ be a graph on $n$ vertices. For $i\in \{0,1\}$ and a connected graph $G$, a spanning forest $F$ of $G$ is called an $i$-perfect forest if every tree in $F$ is an induced subgraph of $G$ and exactly $i$ vertices of $F$ have even degree (including zero). A $i$-perfect forest of $G$ is proper if it has no vertices of degree zero. Scott (2001)...

User authorization queries in the context of role-based access control have attracted considerable interest in the last 15 years. Such queries are used to determine whether it is possible to allocate a set of roles to a user that enables the user to complete a task, in the sense that all the permissions required to complete the task are assigned to...

While there have been many results on lower bounds for Max Cut in unweighted graphs, the only lower bound for non-integer weights is that by Poljak and Turzik (1986). In this paper, we launch an extensive study of lower bounds for Max Cut in weighted graphs. We introduce a new approach for obtaining lower bounds for Weighted Max Cut. Using it, Prob...

Golovach, Paulusma, and Song [Inform. and Comput., 237 (2014), pp. 204--214] asked to determine the parameterized complexity of the following problems parameterized by $k$: 1. Given a graph $G$, a clique modulator $D$ (a clique modulator is a set of vertices, whose removal results in a clique) of size $k$ for $G$, and a list $L(v)$ of colors for ev...

Let D be a strongly connected digraph. The average distance σ̄(v) of a vertex v of D is the arithmetic mean of the distances from v to all other vertices of D. The remoteness ρ(D) and proximity π(D) of D are the maximum and the minimum of the average distances of the vertices of D, respectively. We obtain sharp upper and lower bounds on π(D) and ρ(...

Abasi et al. (2014) introduced the following two problems. In the r -S imple k -P ath problem, given a digraph G on n vertices and positive integers r , k , decide whether G has an r -simple k -path, which is a walk where every vertex occurs at most r times and the total number of vertex occurrences is k . In the ( r , k )-M onomial D etection prob...

Bang-Jensen, Bessy, Havet and Yeo showed that every digraph of independence number at most $2$ and arc-connectivity at least $2$ has an out-branching $B^+$ and an in-branching $B^-$ which are arc-disjoint (such two branchings are called a {\it good pair}), which settled a conjecture of Thomassen for digraphs of independence number $2$. They also pr...

We disprove a conjecture of Petrovic (Graphs $\&$ Combinatorics, 2019) on 4-kings in bipartite hypertournaments and provide a sufficient condition for the conclusion of the conjecture to hold.

We introduce and study two natural generalizations of the Connected VertexCover (VC) problem: the $p$-Edge-Connected and $p$-Vertex-Connected VC problem (where $p \geq 2$ is a fixed integer). Like Connected VC, both new VC problems are FPT, but do not admit a polynomial kernel unless $NP \subseteq coNP/poly$, which is highly unlikely. We prove howe...

Graph routing problems have been investigated extensively in operations research, computer science and engineering due to their ubiquity and vast applications. In this paper, we study constant approximation algorithms for some variations of the cluster general routing problem. In this problem, we are given an edge-weighted complete undirected graph...

Gerke et al. (2019) introduced Netflix Games and proved that every such game has a pure strategy Nash equilibrium. In this paper, we explore the uniqueness of pure strategy Nash equilibria in Netflix Games. Let be a graph and a function, and call the pair a weighted graph. A spanning subgraph H of is called a DP-Nash subgraph if H is bipartite with...

A directed graph $D$ is semicomplete if for every pair $x,y$ of vertices of $D,$ there is at least one arc between $x$ and $y.$ \viol{Thus, a tournament is a semicomplete digraph.} In the Directed Component Order Connectivity (DCOC) problem, given a digraph $D=(V,A)$ and a pair of natural numbers $k$ and $\ell$, we are to decide whether there is a...

Graph routing problems have been investigated extensively in operations research, computer science and engineering due to their ubiquity and vast applications. In this paper, we study constant approximation algorithms for some variations of the general cluster routing problem. In this problem, we are given an edge-weighted complete undirected graph...

A digraph D with n vertices is Hamiltonian (pancyclic and vertex‐pancyclic, respectively) if D contains a Hamilton cycle (a cycle of every length 3,4,…,n, for every vertex v∈V(D), a cycle of every length 3,4,…,n through v, respectively.) It is well‐known that a strongly connected tournament is Hamiltonian, pancyclic, and vertex pancyclic. A digraph...

How does social distancing affect the reach of an epidemic in social networks? We present Monte Carlo simulation results of a Susceptible- Infected-Removed (SIR) model on a network, where individuals are limited in the number of other people they can interact with. While increased social distancing always reduces the spread of an infectious disease...

A digraph D=(V,A) has a good pair at a vertex r if D has a pair of arc-disjoint in- and out-branchings rooted at r. Let T be a digraph with t vertices u1,…,ut and let H1,…Ht be digraphs such that Hi has vertices ui,ji,1≤ji≤ni. Then the composition Q=T[H1,…,Ht] is a digraph with vertex set {ui,ji∣1≤i≤t,1≤ji≤ni} and arc set A(Q)=∪i=1tA(Hi)∪{uijiupqp∣...

A weighted proper orientation of a given graph $G$, denoted by $(D,w)$, is an orientation $D$ with a weight function $w: E(G)\rightarrow \mathbb{Z}_+$, such that the in-weight of any adjacent vertices are distinct, where the in-weight of $v$ in $D$, denoted by $w^-_D(v)$, is the sum of the weights of arcs towards $v$. The weighted proper orientatio...

A strong arc decomposition of a digraph D=(V,A) is a decomposition of its arc set A into two disjoint subsets A1 and A2 such that both of the spanning subdigraphs D1=(V,A1) and D2=(V,A2) are strong. Let T be a digraph with t vertices u1,…,ut and let H1,…,Ht be digraphs such that Hi has vertices ui,ji,1≤ji≤ni. Then the composition Q=T[H1,…,Ht] is a...

An orientation of G is a digraph obtained from G by replacing each edge by exactly one of two possible arcs with the same endpoints. We call an orientation proper if neighboring vertices have different in‐degrees. The proper orientation number of a graph G, denoted by χ→(G), is the minimum maximum in‐degree of a proper orientation of G. Araujo et a...

We explore the uniqueness of pure strategy Nash equilibria in the Netflix Games of Gerke et al. (arXiv:1905.01693, 2019). Let $G=(V,E)$ be a graph and $\kappa:\ V\to \mathbb{Z}_{\ge 0}$ a function, and call the pair $(G, \kappa)$ a capacitated graph. A spanning subgraph $H$ of $(G, \kappa)$ is called a $DP$-Nash subgraph if $H$ is bipartite with pa...

Golovach, Paulusma and Song (Inf. Comput. 2014) asked to determine the parameterized complexity of the following problems parameterized by k: (1) Given a graph G, a clique modulator D (a clique modulator is a set of vertices, whose removal results in a clique) of size k for G, and a list L(v) of colors for every v ∈ V(G), decide whether G has a pro...

The Workflow Satisfiability Problem (WSP) is a problem of interest in access control of information security. In its simplest form, the problem coincides with the Constraint Satisfiability Problem, where the number of variables is usually much smaller than the number of values. Wang and Li (ACM Trans. Inf. Syst. Secur. 2010) were the first to study...

Let $D$ be a strongly connected digraph. The average distance $\bar{\sigma}(v)$ of a vertex $v$ of $D$ is the arithmetic mean of the distances from $v$ to all other vertices of $D$. The remoteness $\rho(D)$ and proximity $\pi(D)$ of $D$ are the maximum and the minimum of the average distances of the vertices of $D$, respectively. We obtain sharp up...

The modern integrated circuit is one of the most complex products engineered to date. It continues to grow in complexity as years progress. As a result, very large-scale integrated (VLSI) circuit design now involves massive design teams employing state-of-the-art computer-aided design (CAD) tools. One of the oldest, yet most important CAD problems...

A digraph D=(V,A) has a good decomposition if A has two disjoint sets A1 and A2 such that both (V,A1) and (V,A2) are strong. Let T be a digraph with vertices u1,…,ut (t≥2) and let H1,…Ht be digraphs such that Hi has vertices ui,ji,1≤ji≤ni. Then the composition Q=T[H1,…,Ht] is a digraph with vertex set {ui,ji:1≤i≤t,1≤ji≤ni} and arc set A(Q)=∪i=1tA(H...

We introduce an extension of decision problems called resiliency problems. In a resiliency problem, the goal is to decide whether an instance remains positive after any (appropriately defined) perturbation has been applied to it. To tackle these kinds of problems, some of which might be of practical interest, we introduce a notion of resiliency for...

Golovach, Paulusma and Song (Inf. Comput. 2014) asked to determine the parameterized complexity of the following problems parameterized by $k$: (1) Given a graph $G$, a clique modulator $D$ (a clique modulator is a set of vertices, whose removal results in a clique) of size $k$ for $G$, and a list $L(v)$ of colors for every $v\in V(G)$, decide whet...

An orientation of $G$ is a digraph obtained from $G$ by replacing each edge by exactly one of two possible arcs with the same endpoints. We call an orientation \emph{proper} if neighbouring vertices have different in-degrees. The proper orientation number of a graph $G$, denoted by $\vec{\chi}(G)$, is the minimum maximum in-degree of a proper orien...