Partial 2-trees Research Articles

On the complexity of the storyplan problem

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.

Drawing Partial 2-Trees with Few Slopes

Drawing Partial 2-Trees with Few Slopes

On mixed linear layouts of series-parallel graphs

A mixed s-stack q-queue layout of a graph consists of a linear order of its vertices and a partition of its edges into s stacks and q queues, such that no two edges in the same stack cross and no two edges in the same queue nest. In 1992, Heath and Rosenberg conjectured that every planar graph admits a mixed 1-stack 1-queue layout. In 2017, Pupyrev disproved this conjecture by demonstrating a planar partial 3-tree that does not admit a mixed 1-stack 1-queue layout.In this work, we strengthen Pupyrev's result by showing that the conjecture does not hold even for 2-trees, also known as series-parallel graphs. We conclude this by means of a stronger result, stating that the conjecture does not hold, even in the more general union and local settings. In the former, crossings (nestings) are allowed in the stack (queue) as long as the involved edges belong to different connected components of the subgraph induced by its edges, while in the latter several stacks and queues may form the layout, but locally every vertex is incident to edges from only one stack and one queue.

On Mixed Linear Layouts of Series-Parallel Graphs

A mixed s-stack q-queue layout of a graph consists of a linear order of its vertices and of a partition of its edges into s stacks and q queues, such that no two edges in the same stack cross and no two edges in the same queue nest. In 1992, Heath and Rosenberg conjectured that every planar graph admits a mixed 1-stack 1-queue layout. Recently, Pupyrev disproved this conjectured by demonstrating a planar partial 3-tree that does not admit a 1-stack 1-queue layout. In this note, we strengthen Pupyrev's result by showing that the conjecture does not hold even for 2-trees, also known as series-parallel graphs.

Edge clique partition in [formula omitted]-graphs

An edge clique partition of G is a set of edge-disjoint cliques of G which together contain every edge of G. In this work we investigate the computational complexity of determining the minimum size of an edge clique partition of G, denoted by cp(G). This problem is NP-hard for split graphs and K4-free graphs. We establish a complete complexity classification of this problem on the hierarchy of (k,ℓ)-graphs, graphs that can be partitioned into k independent sets and ℓ complete sets. In addition, we determine in polynomial time the value of cp(G) for interesting subclasses of (k,ℓ)-graphs: split-indifference graphs, partial 2-trees, and triangular grids.

2-colored point-set embeddings of partial 2-trees

Let G be a planar graph whose vertices are colored either red or blue and let S be a set of points having as many red (resp. blue) points as the red (resp. blue) vertices of G. A 2-colored point-set embedding of G on S is a planar drawing that maps each red (resp. blue) vertex of G to a red (resp. blue) point of S. We show that there exist partial 2-trees that are properly 2-colored (i.e., they are 2-colored with no two adjacent vertices have the same color), whose point-set embeddings may require linearly many bends on linearly many edges. For a contrast, we show that two bends per edge are sufficient for 2-colored point-set embedding of properly 2-colored outerplanar graphs. For separable point sets this bound reduces to one, which is worst-case optimal. If the 2-coloring of the outerplanar graph is not proper, three bends per edge are sufficient and one bend per edge (which is worst-case optimal) is sufficient for caterpillars.

A note on integral generalized flows in directed partial 2-trees

In 2013 Krumke and Zeck developed a pseudopolynomial algorithm to compute maximal integral generalized flows in 2-terminal series-parallel digraphs. In this note we observe that this approach can be extended to the more general class of k-terminal directed partial 2-trees.

Homomorphisms of partial t-trees and edge-colorings of partial 3-trees

A reformulation of the four color theorem is to say that K4 is the smallest graph to which every planar (loop-free) graph admits a homomorphism. Extending this theorem, the second author has proved (using the four color theorem) that the Clebsch graph (a well known triangle-free graph on 16 vertices) is a smallest graph to which every triangle-free planar graph admits a homomorphism. As a further generalization he has proposed that the projective cube of dimension 2k, PC(2k), (that is the Cayley graph (Z22k,{e1,e2,…,e2k,J}, where the ei's are the standard basis and J=e1+e2+⋯+e2k) is a smallest graph of odd-girth 2k+1 to which every planar graph of odd-girth at least 2k+1 admits a homomorphism. This conjecture is related to a conjecture of P. Seymour who claims that the fractional edge-chromatic number of a planar multigraph determines its edge-chromatic number (more precisely, Seymour conjectured that χ′(G)=⌈χf′(G)⌉ for any planar multigraph G). Note that the restriction of Seymour's conjecture to cubic (planar) graphs is Tait's reformulation of the four color theorem.Both these conjectures are believed to be true for the larger class of K5-minor-free graphs (which includes the class of planar graphs). Motivated by these conjectures and in extension of a recent work of L. Beaudou, F. Foucaud and the second author, which studies homomorphism bounds for the class of K4-minor-free graphs, in this work we first give a necessary and sufficient condition for a graph B of odd-girth 2k+1 to admit a homomorphism from any partial t-tree of odd-girth at least 2k+1. Applying our results to the class of partial 3-trees, which is a rich subclass of K5-minor-free graphs, we prove that PC(2k) is in fact a smallest graph of odd-girth 2k+1 to which every partial 3-tree of odd-girth at least 2k+1 admits a homomorphism. We then apply this result to show that every planar (2k+1)-regular multigraph G whose dual is a partial 3-tree, and whose fractional edge-chromatic number is 2k+1, is (2k+1)-edge-colorable. Both these results are the best known supports for the general cases of the above mentioned conjectures in extension of the four color theorem.

NodeTrix Planarity Testing with Small Clusters

We study the NodeTrix planarity testing problem for flat clustered graphs when the maximum size of each cluster is bounded by a constant $k$. We consider both the case when the sides of the matrices to which the edges are incident are fixed and the case when they can be chosen arbitrarily. We show that NodeTrix planarity testing with fixed sides can be solved in $O(k^{3k+\frac{3}{2}} \cdot n)$ time for every flat clustered graph that can be reduced to a partial 2-tree by collapsing its clusters into single vertices. In the general case, NodeTrix planarity testing with fixed sides can be solved in $O(n)$ time for $k = 2$, but it is NP-complete for any $k > 2$. NodeTrix planarity testing remains NP-complete also in the free sides model when $k > 4$.

On maximum common subgraph problems in series–parallel graphs

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 hand, the problem is known to be NP-hard in vertex-labeled partial 11-trees of bounded degree. We consider series–parallel graphs, i.e., partial 2-trees. We show that the problem remains NP-hard in biconnected series–parallel graphs with all but one vertex of degree 3 or less. A positive complexity result is presented for a related problem of high practical relevance which asks for a maximum common connected subgraph that preserves blocks and bridges of the input graphs. We present a polynomial time algorithm for this problem in series–parallel graphs, which utilizes a combination of BC- and SP-tree data structures to decompose both graphs.

Formula omitted]-page and [formula omitted]-page drawings with bounded number of crossings per edge

A drawing of a graph such that the vertices are drawn as points along a line and each edge is a circular arc in one of the two half-planes defined by this line is called a 2-page drawing. If all edges are in the same half-plane, the drawing is called a 1-page drawing. We want to compute 1-page and 2-page drawings of planar graphs such that the number of crossings per edge does not depend on the number of vertices. We show that for any constant k, there exist planar graphs that require more than k crossings per edge in both 1-page and 2-page drawings. We then prove that if the vertex degree is bounded by Δ, every planar 3-tree has a 2-page drawing with a number of crossings per edge that only depends on Δ. Finally, we show a similar result for 1-page drawings of partial 2-trees.

Nonempty intersection of longest paths in series–parallel graphs

In 1966 Gallai asked whether all longest paths in a connected graph have nonempty intersection. This is not true in general and various counterexamples have been found. However, the answer to Gallai’s question is positive for several well-known classes of graphs, as for instance connected outerplanar graphs, connected split graphs, and 2-trees. A graph is series–parallel if it does not contain K4 as a minor. Series–parallel graphs are also known as partial 2-trees, which are arbitrary subgraphs of 2-trees. We present two independent proofs that every connected series–parallel graph has a vertex that is common to all of its longest paths. Since 2-trees are maximal series–parallel graphs, and outerplanar graphs are also series–parallel, our result captures these two classes in one proof and strengthens them to a larger class of graphs. We also describe how one such vertex can be found in linear time.

Drawing Outer 1-planar Graphs with Few Slopes

A graph is outer 1-planar if it admits a drawing where each vertex is on the outer face and each edge is crossed by at most another edge. Outer 1-planar graphs are a superclass of the outerplanar graphs and a subclass of the planar partial 3-trees. We show that an outer 1-planar graph G of bounded degree ∆ admits an outer 1-planar straight-line drawing that uses O(∆) different slopes, which generalizes a previous result by Knauer et al. about the outerplanar slope number of outerplanar graphs (Knauer, Micek, and Walczak. CGTA, 2014). We also show that O(∆2) slopes suffice to construct a crossing-free straight-line drawing of G; the best known upper bound on the planar slope number of planar partial 3-trees of bounded degree ∆ is O(∆5) as proved by Jelínek et al. (V. Jelínek, E. Jelínková, J. Kratochvíl, B. Lidický, M. Tesar, and T. Vyskocil. Graphs and Combinatorics, 2013).

Planar Induced Subgraphs of Sparse Graphs

We show that every graph has an induced pseudoforest of at least n−m/4.5 vertices, an induced partial 2-tree of at least n−m/5 vertices, and an induced planar subgraph of at least n−m/5.2174 vertices. These results are constructive, implying linear-time algorithms to find the respective induced subgraphs. We also show that the size of the largest Kh-minor-free graph in a given graph can sometimes be at most n−m/6+o(m).

Characterization of minimum cycle basis in weighted partial 2-trees

For a weighted outerplanar graph, the set of lex short cycles is known to be a minimum cycle basis (Liu and Lu, 2010). In our work, we show that the set of lex short cycles is a minimum cycle basis in weighted partial 2-trees (graphs of treewidth at most two), which is a superclass of outerplanar graphs. In general graphs, a minimum cycle basis is a subset of the set of lex short cycles, where as the equality is known to hold only for the weighted outerplanar graphs.

Outerplanar graph drawings with few slopes

We consider straight-line outerplanar drawings of outerplanar graphs in which a small number of distinct edge slopes are used, that is, the segments representing edges are parallel to a small number of directions. We prove that Δ−1 edge slopes suffice for every outerplanar graph with maximum degree Δ⩾4. This improves on the previous bound of O(Δ5), which was shown for planar partial 3-trees, a superclass of outerplanar graphs. The bound is tight: for every Δ⩾4 there is an outerplanar graph with maximum degree Δ that requires at least Δ−1 distinct edge slopes in an outerplanar straight-line drawing.

Computing Minimum Cycle Bases in Weighted Partial 2-Trees in Linear Time

We present a linear time algorithm for computing an implicit linear space representation of a minimum cycle basis (MCB) in weighted partial 2-trees, i.e., graphs of treewidth two. The implicit representation can be made explicit in a running time that is proportional to the size of the MCB.

The strong game colouring number of directed graphs

In this paper, we study the strong game colouring number of directed graphs. It is proved that every oriented partial 2-tree has strong game colouring number at most 7 and every oriented planar graph has strong game colouring number at most 15.

Drawing planar 3-trees with given face areas

We study straight-line drawings of planar graphs such that each interior face has a prescribed area. It was known that such drawings exist for all planar graphs with maximum degree 3. We show here that such drawings exist for all planar partial 3-trees, i.e., subgraphs of a triangulated planar graph obtained by repeatedly inserting a vertex in one triangle and connecting it to all vertices of the triangle. Moreover, vertices have rational coordinates if the face areas are rational, and we can bound the resolution. We also give some negative results for other graph classes.

The game Grundy number of graphs

Given a graph G=(V,E), two players, Alice and Bob, alternate their turns in choosing uncoloured vertices to be coloured. Whenever an uncoloured vertex is chosen, it is coloured by the least positive integer not used by any of its coloured neighbours. Alice’s goal is to minimise the total number of colours used in the game, and Bob’s goal is to maximise it. The game Grundy number of G is the number of colours used in the game when both players use optimal strategies. It is proved in this paper that the maximum game Grundy number of forests is 3, and the game Grundy number of any partial 2-tree is at most 7.