Abstract We introduce C lusterix , a novel hybrid visualization model for representing hierarchically clustered networks, which also supports directed and weighted edges. C lusterix offers an integrated view of both the network and its full cluster hierarchy by compactly visualizing the cluster inclusion tree enriched with links of the network. This is achieved through matrix‐based representations at various hierarchy levels, combined with a node‐link style linear layout at the leaf level. To support layout computation based on C lusterix , we propose two algorithmic approaches: an exact Integer Linear Program and a fast heuristic, both aimed at minimizing edge crossings. We present an extensive experimental comparison of these algorithmic approaches to highlight the trade‐offs between efficiency and effectiveness. Moreover, as a proof of concept for our model, we developed an interactive visualization system based on C lusterix and evaluated its performance through case studies and qualitative feedback from experts in different application domains.

A linear layout of a graph defines a total order of the vertices and partitions the edges into either stacks or queues, i.e., crossing-free and non-nested sets of edges along the order, respectively. In this work, we study defective linear layouts that allow forbidden patterns among edges of the same set. Our focus is on k-defective stack layouts and k-defective queue layouts, in which the conflict graph representing the forbidden patterns among the edges of each stack or queue has maximum degree at most k.

We address the problem of computing a dynamic visualization of a geometric graph G as a sequence of frames. Each frame shows only a portion of the graph but their union covers G entirely. The two main requirements of our dynamic visualization are: (i) guaranteeing drawing stability, so to preserve the user’s mental map; (ii) keeping the visual complexity of each frame low. To satisfy the first requirement, we never change the position of the vertices. Regarding the second requirement, we avoid edge crossings in each frame. More precisely, in the first frame we visualize a suitable subset of non-crossing edges; in each subsequent frame, exactly one new edge enters the visualization and all the edges that cross with it are deleted. We call such a sequence of frames a planar story of G. Our goal is to find a planar story whose minimum number of edges contemporarily displayed is maximized (i.e., a planar story that maximizes the minimum frame size). Besides studying our model from a theoretical point of view, we also design and experimentally compare different algorithms, both exact techniques and heuristics. These algorithms provide an array of alternative trade-offs between efficiency and effectiveness, also depending on the structure of the input graph.

We study the problem of representing a graph as a storyplan, a recently introduced model for dynamic graph visualization. It is based on a sequence of frames, each showing a subset of vertices and a planar drawing of their induced subgraphs, where vertices appear and disappear over time. Namely, in the StoryPlan problem, we are given a graph and we want to decide whether there exists a total vertex appearance order for which a storyplan exists. We prove that the problem is NP-complete, and complement this hardness with two parameterized algorithms, one in the vertex cover number and one in the feedback edge set number of the input graph. We prove that partial 3-trees always admit a storyplan, which can be computed in linear time. Finally, we show that the problem remains NP-complete if the vertex appearance order is given and we have to choose how to draw the frames.

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 of the input graph such that the resulting digraph is acyclic, and it contains a single source $s$ and a single sink $t$. Computing an $st$-orientation of a graph can be done efficiently, and it finds notable applications in graph algorithms and in particular in graph drawing. On the other hand, finding an $st$-orientation with at most $k$ transitive edges is more challenging and it was recently proven to be NP-hard already when $k=0$. We strengthen this result by showing that the problem remains NP-hard even for graphs of bounded diameter, and for graphs of bounded vertex degree. These computational lower bounds naturally raise the question about which structural parameters can lead to tractable parameterizations of the problem. Our main result is a fixed-parameter tractable algorithm parameterized by treewidth.

Abstract A k-page upward book embedding (kUBE) of a directed acyclic graph G is a book embeddings of G on k pages with the additional requirement that the vertices appear in a topological ordering along the spine of the book. The kUBE Testing problem, which asks whether a graph admits a kUBE, was introduced in 1999 by Heath, Pemmaraju, and Trenk (SIAM J Comput 28(4), 1999). In a companion paper, Heath and Pemmaraju (SIAM J Comput 28(5), 1999) proved that the problem is linear-time solvable for $$k=1$$ k = 1 and NP-complete for $$k = 6$$ k = 6 . Closing this gap has been a central question in algorithmic graph theory since then. In this paper, we make a major contribution towards a definitive answer to the above question by showing that kUBE Testing is NP-complete for $$k\ge 3$$ k ≥ 3 , even for st-graphs, i.e., acyclic directed graphs with a single source and a single sink. Indeed, our result, together with a recent work of Bekos et al. (Theor Comput Sci 946, 2023) that proves the NP-completeness of 2UBE for planar st-graphs, closes the question about the complexity of the kUBE problem for any k. Motivated by this hardness result, we then focus on the 2UBE Testing for planar st-graphs. On the algorithmic side, we present an $$O(f(\beta )\cdot n+n^3)$$ O ( f ( β ) · n + n 3 ) -time algorithm for 2UBE Testing, where $$\beta $$ β is the branchwidth of the input graph and f is a singly-exponential function on $$\beta $$ β . Since the treewidth and the branchwidth of a graph are within a constant factor from each other, this result immediately yields an FPT algorithm for st-graphs of bounded treewidth. Furthermore, we describe an O(n)-time algorithm to test whether a plane st-graph whose faces have a special structure admits a 2UBE that additionally preserves the plane embedding of the input st-graph. On the combinatorial side, we present two notable families of plane st-graphs that always admit an embedding-preserving $$2$$ 2 UBE.

Abstract We study how to place arrow heads in directed graph drawings aiming at minimizing their overlaps and avoiding intersections between arrow heads and edges. The objective is to support users to correctly and quickly recognize edge orientations, i.e. to deduce unambiguously the edge orientations. Our contribution is two‐fold: (i) We present exact and heuristic algorithms for this arrow placement problem, along with an extensive experimental analysis of these techniques; and (ii) we report on a user study aimed to understand the impact of different arrow placement strategies on performing global and local analysis tasks on directed graph layouts.

We study $k$-page upward book embeddings ($k$UBEs) of $st$-graphs, that is, book embeddings of single-source single-sink directed acyclic graphs on $k$ pages with the additional requirement that the vertices of the graph appear in a topological ordering along the spine of the book. We show that testing whether a graph admits a $k$UBE is NP-complete for $k\geq 3$. A hardness result for this problem was previously known only for $k = 6$ [Heath and Pemmaraju, 1999]. Motivated by this negative result, we focus our attention on $k=2$. On the algorithmic side, we present polynomial-time algorithms for testing the existence of $2$UBEs of planar $st$-graphs with branchwidth $β$ and of plane $st$-graphs whose faces have a special structure. These algorithms run in $O(f(β)\cdot n+n^3)$ time and $O(n)$ time, respectively, where $f$ is a singly-exponential function on $β$. Moreover, on the combinatorial side, we present two notable families of plane $st$-graphs that always admit an embedding-preserving $2$UBE.

A mixed graph has both directed and undirected edges. We study how to compute a crossing-free drawing of an embedded planar mixed graph, such that it is upward ‘as much as possible’. Roughly speaking, in an upward drawing of a mixed graph all (undirected) edges are monotone in the vertical direction and directed edges flow monotonically from bottom to top according to their orientation. We study quasi-upward drawings of mixed graphs, that is, upward drawings where edges can break the vertical monotonicity in a finite number of edge points, called bends. We describe both efficient heuristic techniques and exact approaches for computing quasi-upward planar drawings of embedded mixed graphs with few bends, and we extensively compare them experimentally: the results suggest that our algorithms are effective in many cases.