Optimization problems consist of either maximizing or minimizing an objective function. Instead of looking for a maximum solution (resp. minimum solution), one can find a minimum maximal solution (resp. maximum minimal solution). Such "flipping" of the objective function was done for many classical optimization problems. For example, ${\rm M{\small INIMUM}}$ ${\rm V{\small ERTEX}}$ ${\rm C{\small OVER}}$ becomes ${\rm M{\small AXIMUM}}$ ${\rm M{\small INIMAL}}$ ${\rm V{\small ERTEX}}$ ${\rm C{\small OVER}}$, ${\rm M{\small AXIMUM}}$ ${\rm I{\small NDEPENDENT}}$ ${\rm S{\small ET}}$ becomes ${\rm M{\small INIMUM}}$ ${\rm M{\small AXIMAL}}$ ${\rm I{\small NDEPENDENT}}$ ${\rm S{\small ET}}$ and so on. In this paper, we propose to study the weighted version of Maximum Minimal Edge Cover called ${\rm U{\small PPER}}$ ${\rm E{\small DGE}}$ ${\rm C{\small OVER}}$, a problem having application in genomic sequence alignment. It is well-known that ${\rm M{\small INIMUM}}$ ${\rm E{\small DGE}}$ ${\rm C{\small OVER}}$ is polynomial-time solvable and the "flipped" version is NP-hard, but constant approximable. We show that the weighted ${\rm U{\small PPER}}$ ${\rm E{\small DGE}}$ ${\rm C{\small OVER}}$ is much more difficult than ${\rm U{\small PPER}}$ ${\rm E{\small DGE}}$ ${\rm C{\small OVER}}$ because it is not $O(\frac{1}{n^{1/2-\varepsilon}})$ approximable, nor $O(\frac{1}{\Delta^{1-\varepsilon}})$ in edge-weighted graphs of size $n$ and maximum degree $\Delta$ respectively. Indeed, we give some hardness of approximation results for some special restricted graph classes such as bipartite graphs, split graphs and $k$-trees. We counter-balance these negative results by giving some positive approximation results in specific graph classes.

${\rm F{\small IXED-}F{\small LOOD-}I{\small T}}$ and ${\rm F{\small REE-}F{\small LOOD-}I{\small T}}$ are combinatorial problems on graphs that generalize a very popular puzzle called Flood-It. Both problems consist of recoloring moves whose goal is to produce a monochromatic ("flooded") graph as quickly as possible. Their difference is that in ${\rm F{\small REE-}F{\small LOOD-}I{\small T}}$ the player has the additional freedom of choosing the vertex to play in each move. In this paper, we investigate how this freedom affects the complexity of the problem. It turns out that the freedom is bad in some sense. We show that some cases trivially solvable for ${\rm F{\small IXED-}F{\small LOOD-}I{\small T}}$ become intractable for ${\rm F{\small REE-}F{\small LOOD-}I{\small T}}$. We also show that some tractable cases for ${\rm F{\small IXED-}F{\small LOOD-}I{\small T}}$ are still tractable for ${\rm F{\small REE-}F{\small LOOD-}I{\small T}}$ but need considerably more involved arguments. We finally present some combinatorial properties connecting or separating the two problems. In particular, we show that the length of an optimal solution for ${\rm F{\small IXED-}F{\small LOOD-}I{\small T}}$ is always at most twice that of ${\rm F{\small REE-}F{\small LOOD-}I{\small T}}$, and this is tight.

We show that the eccentricities of all vertices of a $\delta$-hyperbolic graph $G=(V,E)$ can be computed in linear time with an additive one-sided error of at most $c\delta$, i.e., after a linear time preprocessing, for every vertex $v$ of $G$ one can compute in $O(1)$ time an estimate $\hat{e}(v)$ of its eccentricity $ecc_G(v):=\max\{d_G(u,v): u\in V\}$ such that $ecc_G(v)\leq \hat{e}(v)\leq ecc_G(v)+ c\delta$ for a small constant $c$. We prove that every $\delta$-hyperbolic graph $G$ has a shortest path tree $T$, constructible in linear time, such that for every vertex $v$ of $G$, $ecc_G(v)\leq ecc_T(v)\leq ecc_G(v)+ c\delta$, where $ecc_T(v):=\max\{d_T(u,v): u\in V\}$. These results are based on an interesting monotonicity property of the eccentricity function of hyperbolic graphs: the closer a vertex is to the center of $G$, the smaller its eccentricity is. We also show that the distance matrix of $G$ with an additive one-sided error of at most $c'\delta$ can be computed in $O(|V|^2\log^2|V|)$ time, where $c'< c$ is a small constant. Recent empirical studies show that many real-world graphs (including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others) have small hyperbolicity. So, we analyze the performance of our algorithms for approximating eccentricities and distance matrix on a number of real-world networks. Our experimental results show that the obtained estimates are even better than the theoretical bounds.

A geometric graph in the plane is angle-monotone of width $\gamma$ if every pair of vertices is connected by an angle-monotone path of width $\gamma$, a path such that the angles of any two edges in the path differ by at most $\gamma$. Angle-monotone graphs have good spanning properties. We prove that every point set in the plane admits an angle-monotone graph of width $90^\circ$, hence with spanning ratio $\sqrt 2$, and a subquadratic number of edges. This answers an open question posed by Dehkordi, Frati and Gudmundsson. We show how to construct, for any point set of size $n$ and any angle $\alpha$, $0 < \alpha < 45^\circ$, an angle-monotone graph of width $(90^\circ+\alpha)$ with $O(\frac{n}{\alpha})$ edges. Furthermore, we give a local routing algorithm to find angle-monotone paths of width $(90^\circ+\alpha)$ in these graphs. The \emph{routing ratio}, which is the ratio of path length to Euclidean distance, is at most $1/\cos(45^\circ + \frac{\alpha}{2})$, i.e., ranging from $\sqrt 2 \approx 1.414$ to $2.613$. For the special case $\alpha = 30^\circ$, we obtain the full-$\Theta_6$-graph and our routing algorithm achieves the known routing ratio 2 while finding angle-monotone paths of width $120^\circ$.

We present a new heuristic method for minimising crossings in a graph. The method is based upon repeatedly solving the so-called star insertion problem in the setting where the combinatorial embedding is fixed, and has several desirable characteristics for practical use. We introduce the method, discuss some aspects of algorithm design for our implementation, and provide some experimental results. The results indicate that our method compares well to existing methods, and also that it is suitable for dense instances.

Finding cohesive subgraphs in a network has been investigated in many network mining applications. Several alternative formulations of cohesive subgraph have been proposed, a notable one of them is $s$-club, which is a subgraph whose diameter is at most $s$. Here we consider a natural variant of the well-known Minimum Clique Cover problem, where we aim to cover a given graph with the minimum number of $s$-clubs, instead of cliques. We study the computational and approximation complexity of this problem, when $s$ is equal to 2 or 3. We show that deciding if there exists a cover of a graph with three $2$-clubs is NP-complete, and that deciding if there exists a cover of a graph with two $3$-clubs is NP-complete. Then, we consider the approximation complexity of covering a graph with the minimum number of $2$-clubs and $3$-clubs. We show that, given a graph $G=(V,E)$ to be covered, covering $G$ with the minimum number of $2$-clubs is not approximable within factor $O(|V|^{1/2 -\varepsilon})$, for any $\varepsilon>0$, and covering $G$ with the minimum number of $3$-clubs is not approximable within factor $O(|V|^{1 -\varepsilon})$, for any $\varepsilon>0$. On the positive side, we give an approximation algorithm of factor $2|V|^{1/2}\log^{3/2} |V|$ for covering a graph with the minimum number of $2$-clubs.

We consider a single allocation hub-and-spoke network design problem which allocates each non-hub node to exactly one of the given hub nodes in order to minimize the total transportation cost. This paper deals with a cycle-star hub network design problem, in which the hubs are located in a cycle. This problem is essentially equivalent to a cycle-metric labeling problem. It is useful in the design of networks in telecommunications and airline transportation systems. We propose a $2(1-1/h)$-approximation algorithm, where $h$ denotes the number of hub nodes. Our algorithm solves a linear relaxation problem and employs a dependent rounding procedure. We analyze our algorithm by approximating a given cycle-metric matrix using a convex combination of Monge matrices.

We study the effects on graph width parameters of planarization, the construction of a planar diagram from a non-planar graph drawing by replacing each crossing with a new vertex. We show that for treewidth, pathwidth, branchwidth, clique-width, and tree-depth there exists a family of $n$-vertex graphs with bounded parameter value, all of whose planarizations have parameter value $\Omega(n)$. However, for bandwidth, cutwidth, and carving width, every graph with bounded parameter value has a planarization of linear size whose parameter value remains bounded. The same is true for the treewidth, pathwidth, and branchwidth of graphs of bounded degree. To show our lower bounds on the width of planarizations, we prove that arrangements of curves with many crossing pairs of curves must generate planar graphs of high width.

We consider the problem of drawing graphs with at most one crossing per edge. These drawings, and the graphs that can be drawn in this way, are called $1$-planar. Finding $1$-planar drawings is known to be ${\mathsf{NP}}$-hard, but we prove that it is fixed-parameter tractable with respect to the vertex cover number, tree-depth, and cyclomatic number. Special cases of these algorithms provide polynomial-time recognition algorithms for $1$-planar split graphs and $1$-planar cographs. However, recognizing $1$-planar graphs remains ${\mathsf{NP}}$-complete for graphs of bounded bandwidth, pathwidth, or treewidth.

We consider clustered planarity with overlapping clusters as introduced by Didimo et al. (Didimo, Giordano, Liotta, JGAA, 2008). It can be deduced from a proof in Athenstädt et al. (J. C. Athenstädt, T. Hartmann, and M. Nöllenburg, GD 2014) that the problem is NP-complete, even if restricted to instances where the underlying graph is 2-connected, the set of clusters is the union of two partitions and each cluster contains at most two connected components while their complements contain at most three connected components. In this paper, we show that clustered planarity with overlapping clusters can be solved in polynomial time if each cluster induces a connected subgraph. It can be solved in linear time if the set of clusters is the union of two partitions of the vertex set such that, for each cluster, both the cluster and its complement induce connected subgraphs.