The last in-tree recognition problem asks whether a given spanning tree can be derived by connecting each vertex with its rightmost left neighbor of some search ordering. In this study, we demonstrate that the last-in-tree recognition problem for Generic Search is NP-complete. We utilize this finding to strengthen a complexity result from order theory. Given a partial order π and a set of triples, the NP-complete intermezzo problem asks for a linear extension of π where each first element of a triple is not between the other two. We show that this problem remains NP-complete even when the Hasse diagram of the partial order forms a tree of bounded height. In contrast, we give an XP-algorithm for the problem when parameterized by the width of the partial order. Furthermore, we show that - under the assumption of the Exponential Time Hypothesis - the running time of this algorithm is asymptotically optimal. LIPIcs, Vol. 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024), pages 22:1-22:18

Abstract Solving problems on graphs dynamically calls for algorithms to function under repeated modifications to the graph and to be more efficient than solving the problem for the whole graph from scratch after each modification. Dynamic algorithms have been considered for several graph properties, for example connectivity, shortest paths and graph recognition. In this paper we present fully dynamic algorithms for the recognition of threshold graphs and chain graphs, which are optimal in the sense that the costs per modification are linear in the number of modified edges. Furthermore, our algorithms also consider the addition and deletion of sets of vertices as well as edges. In the negative case, i.e., where the graph is not a threshold graph or chain graph anymore, our algorithms return a certificate of constant size. Additionally, we present optimal fully dynamic algorithms for the Hamiltonian cycle problem and the Hamiltonian path problem on threshold and chain graphs which return a vertex cutset as certificate for the non-existence of such a path or cycle in the negative case.

Graph searches and the corresponding search trees can exhibit important structural properties and are used in various graph algorithms. The problem of deciding whether a given spanning tree of a graph is a search tree of a particular search on this graph was introduced by Hagerup in 1985, where the author showed that this problem is efficiently solvable for depth first search (DFS) trees and breadth first search (BFS) trees. If one defines such a search tree in the same way as done for BFS, i.e., by connecting every vertex to its first neighbor, then we call this an ${\cal F}$-tree. If, on the other hand, we connect it with its most recently visited neighbor (as in DFS) we call this an ${\cal L}$-tree. In this paper, we consider related search paradigms. We prove that the search tree problem can be solved in polynomial time for ${\cal L}$-trees of lexicographic depth first search, whereas the ${\cal F}$-tree recognition problem is $\mathcal{NP}$-complete for lexicographic breadth first search, lexicographic depth first search, maximum cardinality search, and maximal neighborhood search. Furthermore, we present polynomial results for both types of trees on chordal graphs.

Lexicographic Depth First Search (LexDFS) is a special variant of a Depth First Search (DFS), which was introduced by Corneil and Krueger in 2008. While this search has been used in various applications, in contrast to other graph searches, no general linear time implementation is known to date. In 2014, Köhler and Mouatadid achieved linear running time to compute some special LexDFS orders for cocomparability graphs. In this paper, we present a linear time implementation of LexDFS for chordal graphs. Our algorithm is able to find any LexDFS order for this graph class. To the best of our knowledge this is the first unrestricted linear time implementation of LexDFS on a non-trivial graph class. In the algorithm we use a search tree computed by Lexicographic Breadth First Search (LexBFS).

In this paper we study how graph searching on a cocomparability graph $G$ can be used to produce cocomp orderings (i.e., orderings that are linear extensions of some transitive orientation of $\overline{G}$) that yield simple algorithms for various intractable problems in general. Such techniques have been used to find a simple certifying algorithm for the minimum path cover problem. In particular we present a characterization of the searches that preserve cocomp orderings when used as a “$^+$” sweep. This allows us to present a toolbox of different graph searches and a framework to solve various problems on cocomparability graphs. We illustrate these techniques by describing a very simple certifying algorithm for the maximum independent set problem as well as a simple permutation graph recognition algorithm.

Traditionally, the coordination of multiple traffic signals and the traffic assignment problem in an urban street network are considered as two separate optimization problems. However, it is easy to see that the traffic assignment has an influence on the optimal signal coordination and, vice versa, a change in the signal coordination changes the optimal traffic assignment. In this article, we present a cyclically time‐expanded network and a corresponding mixed integer linear programming formulation for simultaneously optimizing both the coordination of traffic signals and the traffic assignment in an urban street network. Although the new cyclically time‐expanded network provides a model of both traffic and signals close to reality, it still has the advantage of a linear objective function. Using this model, we compute optimized signal coordinations and traffic assignment on real‐world street networks. To evaluate the practical relevance of the computed solutions, we conduct extensive simulation experiments using two established traffic simulation tools that reveal the advantages of our model. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 65(3), 244–261 2015

Traditionally, the coordination of multiple traffic signals and the traffic assignment problem in an urban street network are considered as two separate optimization problems. However, it is easy to see that the traffic assignment has an influence on the optimal signal coordination and, vice versa, a change in the signal coordination changes the optimal traffic assignment. In this paper we present a cyclically time-expanded network and a corresponding mixed integer linear programming formulation for simultaneously optimizing both the coordination of traffic signals and the traffic assignment in an urban street network. Although the new cyclically time-expanded network provides a model of both traffic and signals close to reality, it still has the advantage of a linear objective function. Using this model we compute optimized signal coordinations and traffic assignment on real-world street networks. To evaluate the practical relevance of the computed solutions we conduct extensive simulation experiments using two established traffic simulation tools that reveal the advantages of our model.

For a given number L , an L -length-bounded edge-cut (node-cut, respectively) in a graph G with source s and sink t is a set C of edges (nodes, respectively) such that no s - t -path of length at most L remains in the graph after removing the edges (nodes, respectively) in C . An L -length-bounded flow is a flow that can be decomposed into flow paths of length at most L . In contrast to classical flow theory, we describe instances for which the minimum L -length-bounded edge-cut (node-cut, respectively) is Θ( n 2/3 )-times (Θ(√ n )-times, respectively) larger than the maximum L -length-bounded flow, where n denotes the number of nodes; this is the worst case. We show that the minimum length-bounded cut problem is NP -hard to approximate within a factor of 1.1377 for L ≥ 5 in the case of node-cuts and for L ≥ 4 in the case of edge-cuts. We also describe algorithms with approximation ratio O (min{ L , n/L }) ⊆ O √ n in the node case and O (min { L , n 2 / L 2 ,√ m } ⊆ O 2/3 in the edge case, where m denotes the number of edges. Concerning L -length-bounded flows, we show that in graphs with unit-capacities and general edge lengths it is NP -complete to decide whether there is a fractional length-bounded flow of a given value. We analyze the structure of optimal solutions and present further complexity results.

Abstract Consider the following problem: compute a spanning tree such that the sum of the lengths of its induced fundamental circuits is as small as possible. We motivate why planar square grid graphs are very relevant instances for this problem. In particular, other contributions already showed that the identification of strong lower bounds is highly challenging. Asymptotically, for a graph on n vertices, Alon et al. [SIAM J Comput 24(1995), 78–100] obtained a lower bound of Ω(n log n). We raise the n log n coefficient by a factor of 325. Concerning optimality proofs, the largest grid for which provably optimum solutions were known is 6 × 6, and it was obtained by massive MIP computing power. Here, we present a combinatorial optimality proof even for the 8 × 8 grid. These two results are complemented by new combinatorial lower bounds for the dimensions in which earlier empirical computations were performed, i.e., for up to 10,000 vertices. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009

Asteroidal triple free (AT‐free) graphs have been introduced as a generalization of interval graphs, since interval graphs are exactly the chordal AT‐free graphs. While for interval graphs it is obvious that there is always a linear ordering of the vertices, such that for each triple of independent vertices the middle one intercepts any path between the remaining vertices of the triple, it is not clear that such an ordering exists for AT‐free graphs in general. In this paper we study graphs that are defined by enforcing such an ordering. In particular, we introduce two subfamilies of AT‐free graphs, namely, path orderable graphs and strong asteroid free graphs. Path orderable graphs are defined by a linear ordering of the vertices that is a natural generalization of the ordering that characterizes cocomparability graphs. On the other hand, motivation for the definition of strong asteroid free graphs comes from the fundamental work of Gallai on comparability graphs. We show that cocomparability graphs $\subset$ path orderable graphs $\subset$ strong asteroid free graphs $\subset$ AT‐free graphs. In addition, we settle the recognition question for the two new classes by proving that recognizing path orderable graphs is NP‐complete, whereas the recognition problem for strong asteroid free graphs can be solved in polynomial time.

We present a first exact study on higher‐dimensional packing problems with order constraints. Problems of this type occur naturally in applications such as logistics or computer architecture and can be interpreted as higher‐dimensional generalizations of scheduling problems. Using graph‐theoretic structures to describe feasible solutions, we develop a novel exact branch‐and‐bound algorithm. This extends previous work by Fekete and Schepers; a key tool is a new order‐theoretic characterization of feasible extensions of a partial order to a given complementarity graph that is tailor‐made for use in a branch‐and‐bound environment. The usefulness of our approach is validated by computational results.

Abstract Recently, considerable effort has been spent on showing that Lexicographic Breadth First Search (LBFS) can be used to determine a tight bound on the diameter of graphs from various restricted classes. In this paper, we show that, in some cases, the full power of LBFS is not required and that other variations of Breadth First Search (BFS) suffice. The restricted graph classes that are amenable to this approach all have a small constant upper bound on the maximum‐sized cycle that may appear as an induced subgraph. We show that, on graphs that have no induced cycle of size greater thank, BFS finds an estimate of the diameter that is no worse than diam(G) − ⌊k/2⌋. © 2003 Wiley Periodicals, Inc.

We consider the optimal placement of hardware modules in space and time for FPGA architectures with reconfiguration capabilities, where modules are modeled as three-dimensional boxes in space and time. Using a graph-theoretic characterization of feasible packings, we are able to solve the following problems. (a) Find the minimal execution time of the given problem on an FPGA of fixed size, (b) Find the FPGA of minimal size to accomplish the tasks within a fired time limit. Furthermore, our approach is perfectly suited for the treatment of precedence constraints for the sequence of tasks, which are present in virtually all practical instances. Additional mathematical structures are developed that lead to a powerful framework for completing optimal solutions. The usefulness is illustrated by computational results.

We prove that claw-free graphs, containing an induced dominating path, have a Hamiltonian path, and that 2-connected claw-free graphs, containing an induced doubly dominating cycle or a pair of vertices such that there exist two internally disjoint induced dominating paths connecting them, have a Hamiltonian cycle. As a consequence, we obtain linear time algorithms for both problems if the input is restricted to (claw,net)-free graphs. These graphs enjoy those interesting structural properties.