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This paper formalises the Canadian Traveller problem as a positional two-player game on graphs. We consider two variants depending on whether an edge is blocked. In the locally-informed variant, the traveller learns if an edge is blocked…
The notion of graph covers (also referred to as locally bijective homomorphisms) plays an important role in topological graph theory and has found its computer science applications in models of local computation. For a fixed target graph…
Let G be a finite undirected graph. A vertex dominates itself and all its neighbors in G. A vertex set D is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient…
In the Directed $k$-Chinese Postman Problem ($k$-DCPP), we are given a connected weighted digraph $G$ and asked to find $k$ non-empty closed directed walks covering all arcs of $G$ such that the total weight of the walks is minimum. Gutin,…
The multicut problem is an NP-hard combinatorial optimization problem with diverse applications in fields such as bioinformatics, data mining and computer vision. Graph neural networks have been defined for the multicut problem but can be…
For graph classes $P_1,...,P_k$, Generalized Graph Coloring is the problem of deciding whether the vertex set of a given graph $G$ can be partitioned into subsets $V_1,...,V_k$ so that $V_j$ induces a graph in the class $P_j$…
Let $\Delta(G)$ and $\chi'(G)$ be the maximum degree and chromatic index of a graph $G$, respectively. Appearing in different forms, Gupta\,(1967), Goldberg\,(1973), Andersen\,(1977), and Seymour\,(1979) made the following conjecture: Every…
We consider the problem of partitioning a graph into a non-fixed number of non-overlapping subgraphs of maximum density. The density of a partition is the sum of the densities of the subgraphs, where the density of a subgraph is its average…
The question of whether 3-Coloring can be solved in polynomial-time for the diameter two graphs is a well-known open problem in the area of algorithmic graph theory. We study the problem restricted to graph classes that avoid cycles of…
The problem of Distance Edge Labeling is a variant of Distance Vertex Labeling (also known as $L_{2,1}$ labeling) that has been studied for more than twenty years and has many applications, such as frequency assignment. The Distance Edge…
In a rainbow version of the classical Tur\'an problem one considers multiple graphs on a common vertex set, thinking of each graph as edges in a distinct color, and wants to determine the minimum number of edges in each color which…
An edge-colored directed graph is \emph{observable} if an agent that moves along its edges is able to determine his position in the graph after a sufficiently long observation of the edge colors. When the agent is able to determine his…
We show that given a 3-colorable graph, it is NP-hard to find a 3-coloring with $(16/17 + \eps)$ of the edges bichromatic. In a related result, we show that given a satisfiable instance of the 2-to-1 Label Cover problem, it is NP-hard to…
The multicommodity flow problem is NP-hard already for two commodities over bipartite graphs. Nonetheless, using our recent theory of n-fold integer programming and extensions developed herein, we are able to establish the surprising…
A mixed graph contains (undirected) edges as well as (directed) arcs, thus generalizing undirected and directed graphs. A proper coloring $c$ of a mixed graph $G$ assigns a positive integer to each vertex such that $c(u)\neq c(v)$ for every…
A solution to a problem of Erd\H{o}s, Rubin and Taylor is obtained by showing that if a graph $G$ is $(a:b)$-choosable, and $c/d > a/b$, then $G$ is not necessarily $(c:d)$-choosable. The simplest case of another problem, stated by the same…
We consider the problem of dualizing a monotone CNF (equivalently, computing all minimal transversals of a hypergraph), whose associated decision problem is a prominent open problem in NP-completeness. We present a number of new polynomial…
Let $G=(V,E)$ be a vertex-colored graph, where $C$ is the set of colors used to color $V$. The Graph Motif (or GM) problem takes as input $G$, a multiset $M$ of colors built from $C$, and asks whether there is a subset $S\subseteq V$ such…
Orienting the edges of an undirected graph such that the resulting digraph satisfies some given constraints is a classical problem in graph theory, with multiple algorithmic applications. In particular, an $st$-orientation orients each edge…
A colored graph is a directed graph in which nodes or edges have been assigned colors that are not necessarily unique. Observability problems in such graphs consider whether an agent observing the colors of edges or nodes traversed on a…