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Determining whether two graphs are structurally identical is a fundamental problem with applications spanning mathematics, computer science, chemistry, and network science. Despite decades of study, graph isomorphism remains a challenging…
The modular decomposition is a technique that applies but is not restricted to graphs. The notion of module naturally appears in the proofs of many graph theoretical theorems. Computing the modular decomposition tree is an important…
Motivated by the problem of testing planarity and related properties, we study the problem of designing efficient {\em partition oracles}. A {\em partition oracle} is a procedure that, given access to the incidence lists representation of a…
This work introduces a novel algorithm for finding the connected components of a graph where the vertices and edges are grouped into sets defining a Set--Based Graph. The algorithm, under certain restrictions on those sets, has the…
A geometric intersection graph is constructed over a set of geometric objects, where each vertex represents a distinct object and an edge connects two vertices if and only if the corresponding objects intersect. We examine the problem of…
The clique-width is a measure of complexity of decomposing graphs into certain tree-like structures. The class of graphs with bounded clique-width contains bounded tree-width graphs. We give a polynomial time graph isomorphism algorithm for…
We present a data structure that, given a graph $G$ of $n$ vertices and $m$ edges, and a suitable pair of nested $r$-divisions of $G$, preprocesses $G$ in $O(m+n)$ time and handles any series of edge-deletions in $O(m)$ total time while…
We study how the number $c(X)$ of components of a graph $X$ can be expressed through the number and properties of the components of a quotient graph $X/\sim.$ We partially rely on classic qualifications of graph homomorphisms such as…
In this paper we consider the separability problem for bipartite quantum states arising from graphs. Earlier it was proved that the degree criterion is the graph-theoretic counterpart of the familiar positive partial transpose criterion for…
The K-way vertex cut problem} consists in, given a graph G, finding a subset of vertices of a given size, whose removal partitions G into the maximum number of connected components. This problem has many applications in several areas. It…
There is a long line of research in the literature dedicated to word-representable graphs, which generalize several important classes of graphs. However, not much is known about word-representability of split graphs, another important class…
Compound graphs are networks in which vertices can be grouped into larger subsets, with these subsets capable of further grouping, resulting in a nesting that can be many levels deep. In several applications, including biological workflows,…
Intersection graphs of planar geometric objects such as intervals, disks, rectangles and pseudo-disks are well studied. Motivated by various applications, Butman et al. in SODA 2007 considered algorithmic questions in intersection graphs of…
In this paper, we propose a new type of graph, denoted as "embedded-graph", and its theory, which employs a distributed representation to describe the relations on the graph edges. Embedded-graphs can express linguistic and complicated…
In this work we discuss some appearances of semi-infinite combinatorics in representation theory. We propose a semi-infinite moment graph theory and we motivate it by considering the (not yet rigorously defined) geometric side of the story.…
Traditional representations of graphs and their duals suggest the requirement that the dual vertices be placed inside their corresponding primal faces, and the edges of the dual graph cross only their corresponding primal edges. We consider…
The complexity of the maximum common connected subgraph problem in partial $k$-trees is still not fully understood. Polynomial-time solutions are known for degree-bounded outerplanar graphs, a subclass of the partial $2$-trees. On the other…
We treat the problem of characterizing in a systematic way the qualitative features of two-dimensional dynamical systems. To that end, we construct a representation of the topological features of phase portraits by means of diagrams that…
A vertex set $U \subseteq V$ of an undirected graph $G=(V,E)$ is a $\textit{resolving set}$ for $G$, if for every two distinct vertices $u,v \in V$ there is a vertex $w \in U$ such that the distances between $u$ and $w$ and the distance…
A graph G is prismatic if for every triangle T of G, every vertex of G not in T has a unique neighbour in T. The complement of a prismatic graph is called \emph{antiprismatic}. The complexity of colouring antiprismatic graphs is still…