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Graph embedding is a powerful method in parallel computing that maps a guest network $G$ into a host network $H$. The performance of an embedding can be evaluated by certain parameters, such as the dilation, the edge congestion and the…
This paper leverages the framework of algorithms-with-predictions to design data structures for two fundamental dynamic graph problems: incremental topological ordering and cycle detection. In these problems, the input is a directed graph…
An \emph{s-graph} is a graph with two kinds of edges: \emph{subdivisible} edges and \emph{real} edges. A \emph{realisation} of an s-graph $B$ is any graph obtained by subdividing subdivisible edges of $B$ into paths of arbitrary length (at…
We study the design of fixed-parameter algorithms for problems already known to be solvable in polynomial time. The main motivation is to get more efficient algorithms for problems with unattractive polynomial running times. Here, we focus…
We consider a natural combinatorial optimization problem on chordal graphs, the class of graphs with no induced cycle of length four or more. A subset of vertices of a chordal graph is (monophonically) convex if it contains the vertices of…
A graph is chordal if every cycle of length at least four contains a chord, that is, an edge connecting two nonconsecutive vertices of the cycle. Several classical applications in sparse linear systems, database management, computer vision,…
The NP-complete problem Feedback Vertex Set is that of deciding whether or not it is possible, for a given integer $k\geq 0$, to delete at most $k$ vertices from a given graph so that what remains is a forest. The variant in which the…
A chordless cycle, or equivalently a hole, in a graph $G$ is an induced subgraph of $G$ which is a cycle of length at least $4$. We prove that the Erd\H{o}s-P\'osa property holds for chordless cycles, which resolves the major open question…
The NP-hard Maximum Planar Subgraph problem asks for a planar subgraph $H$ of a given graph $G$ such that $H$ has maximum edge cardinality. For more than two decades, the only known non-trivial exact algorithm was based on integer linear…
Chordless cycles are very natural structures in undirected graphs, with an important history and distinguished role in graph theory. Motivated also by previous work on the classical problem of listing cycles, we study how to list chordless…
Given an undirected, unweighted graph with $n$ vertices and $m$ edges, the maximum cut problem is to find a partition of the $n$ vertices into disjoint subsets $V_1$ and $V_2$ such that the number of edges between them is as large as…
A long-standing conjecture of Thomassen says that every longest cycle of a $3$-connected graph has a chord. Thomassen (2018) proved that if $G$ is $2$-connected and cubic, then any longest cycle must have a chord. He also showed that if $G$…
For large $n$ we determine the maximum number of induced 6-cycles which can be contained in a planar graph on $n$ vertices, and we classify the graphs which achieve this maximum. In particular we show that the maximum is achieved by the…
Given an Eulerian graph G, in the Maximum Eulerian Cycle Decomposition problem, we are interested in finding a collection of edge-disjoint cycles {E_1, E_2, ..., E_k} in G such that all edges of G are in exactly one cycle and k is maximum.…
The maximum cardinality of an induced $2$-regular subgraph of a graph $G$ is denoted by $c_{\rm ind}(G)$. We prove that if $G$ is an $r$-regular graph of order $n$, then $c_{\rm ind}(G) \geq \frac{n}{2(r-1)} + \frac{1}{(r-1)(r-2)}$ and we…
A linear program with linear complementarity constraints (LPCC) requires the minimization of a linear objective over a set of linear constraints together with additional linear complementarity constraints. This class has emerged as a…
We explore the clearing problem in the barter exchange market. The problem, described in the terminology of graph theory, is to find a set of vertex-disjoint, length-restricted cycles that maximize the total weight in a weighted digraph.…
We study the maximum induced matching problem on a graph g. Induced matchings correspond to independent sets in L2(g), the square of the line graph of g. The problem is NP-complete on bipartite graphs. In this work, we show that for a…
In this paper, we focus on the class of complete $S$-partite graphs, for $S$ an undirected graph possibly with self-loops, and address the problem of finding largest $2$-regular subgraphs of these graphs, which can be formulated as an…
We study a large family of graph covering problems, whose definitions rely on distances, for graphs of bounded cyclomatic number (that is, the minimum number of edges that need to be removed from the graph to destroy all cycles). These…