Complexity aspects of the computation of the rank of a graph

arXiv (Cornell University), 2015

We study convexity properties of graphs. In this paper we present a linear-time algorithm for the geodetic number in tree-cographs. Settling a 10-year-old conjecture, we prove that the Steiner number is at least the geodetic number in AT-free graphs. Computing a maximal and proper monophonic set in AT-free graphs is NP-complete. We present polynomial algorithms for the monophonic number in permutation graphs and the geodetic number in P 4-sparse graphs.

Graphs and Combinatorics, 2012

A set of vertices S in a graph is convex if it contains all vertices which belong to shortest paths between vertices in S. The convexity number c(G) of a graph G is the maximum cardinality of a convex set of vertices which does not contain all vertices of G.

Electronic Notes in Discrete Mathematics, 2011

Given a graph G = (V, E), the closed interval of a pair of vertices u, v ∈ V , denoted by I[u, v], is the set of vertices that belongs to some shortest (u, v)-path. For a given S ⊆ V , let I[S] = u,v∈S I[u, v]. We say that S ⊆ V is a convex set if I[S] = S. The convex hull I h [S] of a subset S ⊆ V is the smallest convex set that contains S. We say that S is a hull set if I h [S] = V. The cardinality of a minimum hull set of G is the hull number of G, denoted by hn(G). We show that deciding if hn(G) ≤ k is an NP-complete problem, even if G is bipartite. We also prove that hn(G) can be computed in polynomial time for cactus and P 4-sparse graphs.

2008

Rankwidth is a graph parameter introduced by Oum and Seymour, based on ranks of adjacency matrices over GF(2). We propose an alternative definition of rankwidth, based on the graph-theoretical notion of H-joins, and give fast dynamic programming algorithms to solve optimization problems on graphs of bounded rankwidth. Such algorithms are interesting since graphs of rankwidth at most k encompass large classes of graphs, e.g., all graphs of treewidth k + 1 or branchwidth k or cliquewidth k , some graphs of unbounded treewidth and branchwidth, and some graphs of cliquewidth 2 k/2−1 − 1. We introduce a graph composition operation called H-join, indexed by a fixed bipartite graph H , and define the H-join decomposable graphs to be those having an H-join decomposition. We show that any problem expressible in MSO1-logic is fixed parameter tractable on an H-join decomposable graph when parameterized by ρ(H) , the rank of the adjacency matrix of H. Given an H-join decomposition of an n-vertex m-edge graph G we solve the Maximum Independent Set and Minimum Dominating Set problems on G in time O(n(m + 2 O(ρ(H) 2))) , and the q-Coloring problem in time O(n(m + 2 O(qρ(H) 2))). For any positive integer k we define a bipartite graph R k and show that the graphs of rankwidth at most k are exactly the R k-join decomposable graphs. Moreover, a rank-decomposition of width k of a graph is also an R k-join decomposition of the graph. For a graph G of rankwidth k , given with its width k rank-decomposition, this results in algorithms which, in O(n(m + 2 1 2 k 2 + 9 2 k × k 2)) time solve the Maximum Independent Set problem on G , in O(n(m + 2 5 4 k 2 + 29 4 k × k 6)) time solve the Minimum Dominating Set problem on G , and in O(n(m + 2 q 4 k 2 + 5q+8 4 k × k × q)) time solve the q-Coloring problem on G. These are the first algorithms for NP-hard problems whose runtimes are less than double exponential in the rankwidth k .

Cornell University - arXiv, 2022

Given a graph G with a terminal set R ⊆ V (G), the Steiner tree problem (STREE) asks for a set S ⊆ V (G) \ R such that the graph induced on S ∪ R is connected. A split graph is a graph which can be partitioned into a clique and an independent set. It is known that STREE is NP-complete on split graphs [1]. To strengthen this result, we introduce convex ordering on one of the partitions (clique or independent set), and prove that STREE is polynomial-time solvable for tree-convex split graphs with convexity on clique (K), whereas STREE is NP-complete on tree-convex split graphs with convexity on independent set (I). We further strengthen our NP-complete result by establishing a dichotomy which says that for unary-tree-convex split graphs (path-convex split graphs), STREE is polynomial-time solvable, and NP-complete for binary-tree-convex split graphs (comb-convex split graphs). We also show that STREE is polynomial-time solvable for triad-convex split graphs with convexity on I, and circular-convex split graphs. Further, we show that STREE can be used as a framework for the dominating set problem (DS) on split graphs, and hence the classical complexity (P vs NPC) of STREE and DS is the same for all these subclasses of split graphs. Furthermore, it is important to highlight that in [2], it is incorrectly claimed that the problem of finding a minimum dominating set on split graphs cannot be approximated within (1 −) ln |V (G)| in polynomial-time for any > 0 unless NP ⊆ DTIME n O(log log n). When the input is restricted to split graphs, we show that the minimum dominating set problem has 2 − 1 |I|-approximation algorithm that runs in polynomial time. Finally, from the parameterized perspective with solution size being the parameter, we show that the Steiner tree problem on split graphs is W [2]-hard, whereas when the parameter is treewidth and the solution size, we show that the problem is fixed-parameter tractable, and if the parameter is the solution size and the maximum degree of I (d), then we show that the Steiner tree problem on split graphs has a kernel of size at most (2d − 1)k d−1 + k, k = |S|.

Electronic Notes in Discrete Mathematics, 2007

A set of vertices S of a graph G is convex if all vertices of every geodesic between two of its vertices are in S. We say that G is k-convex if V (G) can be partitioned into k convex sets. The convex partition number of G is the least k ≥ 2 for which G is k-convex. In this paper we examine k-convexity of graphs. We show that it is NP-complete to decide if G is k-convex, for any fixed k ≥ 2. We describe a characterization for k-convex cographs, leading to a polynomial time algorithm to recognize if a cograph is k-convex. Finally, we discuss k-convexity for disconnected graphs.

Lecture Notes in Computer Science, 2016

A set C of vertices of a graph is P 3-convex if every vertex outside C has at most one neighbor in C. The convex hull σ(A) of a set A is the smallest P 3-convex set that contains A. A set M is convexly independent if for every vertex x ∈ M, x / ∈ σ(M − x). We show that the maximal number of vertices that a convexly independent set in a permutation graph can have, can be computed in polynomial time.

1999

The properties of singular graphs obtained in a previous paper "On the construction of graphs of nullity one", lead to the characterization of graphs of small rank. The minimal conflgurations that are contained in singular graphs were identifled as "grown" from certain cores. A core of a singular graph G is a subgraph induced by the vertices corresponding to the non-zero components of an eigenvector in the nullspace of the adjacency matrix of G. In this paper it is shown that an arbitrary singular graph Z without isolated vertices has core-sizes corresponding to a minimal basis for the nullspace of A bounded below by 2 and above by r(Z) + 1, r(Z) being the rank of Z. For r(Z) µ 6, these bounds are sharp.

Theoretical Computer Science, 2007

Extending previous NP-completeness results for the harmonious coloring problem and the pair-complete coloring problem on trees, bipartite graphs and cographs, we prove that these problems are also NP-complete on connected bipartite permutation graphs. We also study the k-path partition problem and, motivated by a recent work of Steiner [G. Steiner, On the k-path partition of graphs, Theoret. Comput. Sci. 290 (2003Sci. 290 ( ) 2147Sci. 290 ( -2155]], where he left the problem open for the class of convex graphs, we prove that the kpath partition problem is NP-complete on convex graphs. Moreover, we study the complexity of these problems on two well-known subclasses of chordal graphs namely quasi-threshold and threshold graphs. Based on the work of Bodlaender [H.L. Bodlaender, Achromatic number is NP-complete for cographs and interval graphs, Inform. Process. Lett. 31 (1989) 135-138], we show NPcompleteness results for the pair-complete coloring and harmonious coloring problems on quasi-threshold graphs. Concerning the k-path partition problem, we prove that it is also NP-complete on this class of graphs. It is known that both the harmonious coloring problem and the k-path partition problem are polynomially solvable on threshold graphs. We show that the pair-complete coloring problem is also polynomially solvable on threshold graphs by describing a linear-time algorithm.

The Electronic Journal of Combinatorics, 2016

The paper by M. Baker and S. Norine in 2007 introduced a new parameter on configurations of graphs and gave a new result in the theory of graphs which has an algebraic geometry flavor. This result was called Riemann-Roch formula for graphs since it defines a combinatorial version of divisors and their ranks in terms of configurations on graphs. The so called chip firing game on graphs and the sandpile model in physics play a central role in this theory. In this paper we present an algorithm for the determination of the rank of configurations for the complete graph $K_n$. This algorithm has linear arithmetic complexity. The analysis of number of iterations in a less optimized version of this algorithm leads to an apparently new parameter which we call the prerank. This parameter and the parameter dinv provide an alternative description to some well known $q,t$-Catalan numbers. Restricted to a natural subset of configurations, the two natural statistics degree and rank lead to a dist...