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View article: Strong chordality in tournaments and multipartite tournaments with possible loops
Strong chordality in tournaments and multipartite tournaments with possible loops Open
Strongly chordal digraphs are included in the class of chordal digraphs and generalize strongly chordal graphs and chordal bipartite graphs. They are the digraphs that admit a linear ordering of its vertex set for which their adjacency mat…
View article: Trees with minimum weighted Szeged index
Trees with minimum weighted Szeged index Open
The weighted Szeged index is a recent extension of the well-known Szeged index. Trees are conjectured to achieve the minimum weighted Szeged index among all graphs with a given number of vertices. In this paper, we present new tools to ana…
View article: List homomorphisms to separable signed graphs
List homomorphisms to separable signed graphs Open
The complexity of the list homomorphism problem for signed graphs appears difficult to classify. Existing results focus on special classes of signed graphs, such as trees and reflexive signed graphs. Irreflexive signed graphs are in a cert…
View article: Strong cocomparability graphs and Slash-free orderings of matrices
Strong cocomparability graphs and Slash-free orderings of matrices Open
We introduce the class of strong cocomparability graphs, as the class of reflexive graphs whose adjacency matrix can be rearranged by a simultaneous row and column permutation to avoid the submatrix with rows 01, 10, which we call Slash. W…
View article: Min orderings and list homomorphism dichotomies for signed and unsigned graphs
Min orderings and list homomorphism dichotomies for signed and unsigned graphs Open
The CSP dichotomy conjecture has been recently established, but a number of other dichotomy questions remain open, including the dichotomy classification of list homomorphism problems for signed graphs. Signed graphs arise naturally in man…
View article: Describing hereditary properties by forbidden circular orderings
Describing hereditary properties by forbidden circular orderings Open
Each hereditary property can be characterized by its set of minimal obstructions; these sets are often unknown, or known but infinite. By allowing extra structure it is sometimes possible to describe such properties by a finite set of forb…
View article: Describing hereditary properties by forbidden circular orderings
Describing hereditary properties by forbidden circular orderings Open
Each hereditary property can be characterized by its set of minimal obstructions; these sets are often unknown, or known but infinite. By allowing extra structure it is sometimes possible to describe such properties by a finite set of forb…
View article: On the Kernel and Related Problems in Interval Digraphs
On the Kernel and Related Problems in Interval Digraphs Open
Given a digraph G, a set X ⊆ V(G) is said to be an absorbing set (resp. dominating set) if every vertex in the graph is either in X or is an in-neighbour (resp. out-neighbour) of a vertex in X. A set S ⊆ V(G) is said to be an independent s…
View article: Minimum Weighted Szeged Index Trees
Minimum Weighted Szeged Index Trees Open
Weighted Szeged index is a recently introduced extension of the well-known Szeged index. In this paper, we present a new tool to analyze and characterize minimum weighted Szeged index trees. We exhibit the best trees with up to 81 vertices…
View article: List homomorphism problems for signed trees
List homomorphism problems for signed trees Open
We consider homomorphisms of signed graphs from a computational perspective. In particular, we study the list homomorphism problem seeking a homomorphism of an input signed graph $(G,σ)$, equipped with lists $L(v) \subseteq V(H), v \in V(G…
View article: Hamiltonian cycles in covering graphs of trees
Hamiltonian cycles in covering graphs of trees Open
Hamiltonicity of graphs possessing symmetry has been a popular subject of research, with focus on vertex-transitive graphs, and in particular on Cayley graphs. In this paper, we consider the Hamiltonicity of another class of graphs with sy…
View article: Bi-Arc Digraphs and Conservative Polymorphisms *
Bi-Arc Digraphs and Conservative Polymorphisms * Open
In this paper we study the class of bi-arc digraphs, important from two seemingly unrelated perspectives. On the one hand, they are precisely the digraphs that admit certain polymorphisms of interest in the study of constraint satisfaction…
View article: Strongly chordal digraphs and $\Gamma$-free matrices.
Strongly chordal digraphs and $\Gamma$-free matrices. Open
We define strongly chordal digraphs, which generalize strongly chordal graphs\nand chordal bipartite graphs, and are included in the class of chordal\ndigraphs. They correspond to square 0,1 matrices that admit a simultaneous row\nand colu…
View article: Strongly chordal digraphs and $Γ$-free matrices
Strongly chordal digraphs and $Γ$-free matrices Open
We define strongly chordal digraphs, which generalize strongly chordal graphs and chordal bipartite graphs, and are included in the class of chordal digraphs. They correspond to square 0,1 matrices that admit a simultaneous row and column …
View article: Vertex arboricity of cographs
Vertex arboricity of cographs Open
Arboricity is a graph parameter akin to chromatic number, in that it seeks to partition the vertices into the smallest number of sparse subgraphs. Where for the chromatic number we are partitioning the vertices into independent sets, for t…
View article: Complexity of acyclic colorings of graphs and digraphs with degree and girth constraints
Complexity of acyclic colorings of graphs and digraphs with degree and girth constraints Open
We consider acyclic r-colorings in graphs and digraphs: they color the vertices in r colors, each of which induces an acyclic graph or digraph. (This includes the dichromatic number of a digraph, and the arboricity of a graph.) For any gir…
View article: Comparability and Cocomparability Bigraphs
Comparability and Cocomparability Bigraphs Open
We propose bipartite analogues of comparability and cocomparability graphs. Surprizingly, the two classes coincide. We call these bipartite graphs cocomparability bigraphs. We characterize cocomparability bigraphs in terms of vertex orderi…
View article: Distance-Two Colorings of Barnette Graphs
Distance-Two Colorings of Barnette Graphs Open
Barnette identified two interesting classes of cubic polyhedral graphs for which he conjectured the existence of a Hamiltonian cycle. Goodey proved the conjecture for the intersection of the two classes. We examine these classes from the p…
View article: Interval-Like Graphs and Digraphs
Interval-Like Graphs and Digraphs Open
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, also adjusted interval dig…
View article: Complexity of Correspondence Homomorphisms
Complexity of Correspondence Homomorphisms Open
Correspondence homomorphisms are both a generalization of standard homomorphisms and a generalization of correspondence colourings. For a fixed target graph $H$, the problem is to decide whether an input graph $G$, with each edge labeled b…
View article: Minimal obstructions to $2$-polar cographs
Minimal obstructions to $2$-polar cographs Open
A graph is a cograph if it is $P_4$-free. A $k$-polar partition of a graph $G$ is a partition of the set of vertices of $G$ into parts $A$ and $B$ such that the subgraph induced by $A$ is a complete multipartite graph with at most $k$ part…
View article: Minimal digraph obstructions for small matrices
Minimal digraph obstructions for small matrices Open
Given a $\{ 0, 1, \ast \}$-matrix $M$, a minimal $M$-obstruction is a digraph $D$ such that $D$ is not $M$-partitionable, but every proper induced subdigraph of $D$ is. In this note we present a list of all the $M$-obstructions for every $…