Chordal Research Articles

Defective Ramsey Numbers and Defective Cocolorings in Some Subclasses of Perfect Graphs

In this paper, we investigate a variant of Ramsey numbers called defective Ramsey numbers where cliques and independent sets are generalized to k-dense and k-sparse sets, both commonly called k-defective sets. We focus on the computation of defective Ramsey numbers restricted to some subclasses of perfect graphs. Since direct proof techniques are often insufficient for obtaining new values of defective Ramsey numbers, we provide a generic algorithm to compute defective Ramsey numbers in a given target graph class. We combine direct proof techniques with our efficient graph generation algorithm to compute several new defective Ramsey numbers in perfect graphs, bipartite graphs and chordal graphs. We also initiate the study of a related parameter, denoted by \(c^{{\mathcal {G}}}_k(m)\), which is the maximum order n such that the vertex set of any graph of order n in a class \({\mathcal {G}}\) can be partitioned into at most m subsets each of which is k-defective. We obtain several values for \(c^{{\mathcal {G}}}_k(m)\) in perfect graphs and cographs.

Wait-free approximate agreement on graphs

Approximate agreement is one of the few variants of consensus that can be solved in a wait-free manner in asynchronous systems where processes communicate by reading and writing to shared memory. In this work, we consider a natural generalisation of approximate agreement on arbitrary undirected connected graphs. Each process is given a node of the graph as input and, if non-faulty, must output a node such that–all the outputs are within distance 1 of one another, and–each output value lies on a shortest path between two input values. From prior work, it is known that there is no wait-free algorithm among n≥3 processes for this problem on any cycle of length c≥4, by reduction from 2-set agreement (Castañeda et al., 2018).In this work, we investigate the solvability of this task on general graphs. We give a new, direct proof of the impossibility of approximate agreement on cycles of length c≥4, via a generalisation of Sperner's Lemma to convex polygons. We also extend the reduction from 2-set agreement to a larger class of graphs, showing that approximate agreement on these graphs is unsolvable. On the positive side, we present a wait-free algorithm for a different class of graphs, which properly contains the class of chordal graphs.

Algorithmic results in Roman dominating functions on graphs

Given a graph G=(V,E), a subset D⊆V (respectively, function f:V→{0,1,2}) is a dominating set (DS) (respectively, Roman dominating function (RDF)) of G if each vertex v∈V∖D (respectively, v∈V with f(v)=0) is adjacent to a vertex u∈D (respectively, u∈V with f(u)=2). The domination number of G is the minimum cardinality of an DS of G and the Roman domination number of G is the minimum weight of an RDF f of G, where the weight of f is ∑v∈Vf(v). The (Roman) domination problem is to compute the (Roman) domination number of a given graph. In this paper, we study the Roman domination problem. We show that the complexity of the problem differs from the complexity of the domination problem and the problem is NP-complete for circle graphs and undirected path graphs and is APX-complete for graphs of degree at most 4. We also propose an integer linear programming (ILP) formulation with polynomial number of constraints for the problem. Additionally, we use the ILP formulation to give an H(Δ(G)+1)-approximation algorithm for solving the problem for any graph G, where Δ(G) is the maximum degree of G. Furthermore, we show that the optimization version of the problem on split and chordal graphs cannot be approximated in polynomial time within (1/2−ε)ln⁡|V| for any ε>0, unless NP ⊆ DTIME (|V|O(log⁡log⁡|V|)).

Pebbling in powers of paths

The t-fold pebbling number, πt(G), of a graph G is defined to be the minimum number m so that, from any given configuration of m pebbles on the vertices of G, it is possible to place at least t pebbles on any specified vertex via pebbling moves. It has been conjectured that the pebbling numbers of pyramid-free chordal graphs can be calculated in polynomial time.The kth power G(k) of the graph G is obtained from G by adding an edge between any two vertices of distance at most k from each other. The kth power of the path Pn on n vertices is an important class of pyramid-free chordal graphs, and is a stepping stone to the more general class of k-paths and the still more general class of interval graphs. Pachter, Snevily, and Voxman (1995) calculated π(Pn(2)), Kim (2004) calculated π(Pn(3)), and Kim and Kim (2010) calculated π(Pn(4)). In this paper we calculate πt(Pn(k)) for all n, k, and t.For a function D:V(G)→N, the D-pebbling number, π(G,D), of a graph G is defined to be the minimum number m so that, from any given configuration of m pebbles on the vertices of G, it is possible to place at least D(v) pebbles on each vertex v via pebbling moves. It has been conjectured that π(G,D)≤π|D|(G) for all G and D. We make the stronger conjecture that every G and D satisfies π(G,D)≤π|D|(G)−(s(D)−1), where s(D) counts the number of vertices v with D(v)>0. We prove that trees and Pn(k), for all n and k, satisfy the stronger conjecture.The pebbling exponenteπ(G) of a graph G was defined by Pachter et al., to be the minimum k for which π(G(k))=n(G(k)). Of course, eπ(G)≤diam(G), and Czygrinow, Hurlbert, Kierstead, and Trotter (2002) proved that almost all graphs G have eπ(G)=1. Lourdusamy and Mathivanan (2015) proved several results on πt(Cn2), and Hurlbert (2017) proved an asymptotically tight formula for eπ(Cn). Our formula for πt(Pn(k)) allows us to compute eπ(Pn) within additively narrow bounds.

On alpha -Excellent Graphs

A graph G is alpha -excellent if every vertex of G is contained in some maximum independent set of G. In this paper, we characterize alpha -excellent bipartite graphs, alpha -excellent unicyclic graphs, alpha -excellent simplicial graphs, alpha -excellent chordal graphs, alpha -excellent block graphs, and we show that every generalized Petersen graph is alpha -excellent.

Decremental Optimization of Vertex-Coloring Under the Reconfiguration Framework

Suppose that we are given a positive integer k, and a k-(vertex-)coloring of a given graph G. Then we are asked to find a coloring of G using the minimum number of colors among colorings that are reachable from by iteratively changing a color assignment of exactly one vertex while maintaining the property of k-colorings. In this paper, we give linear-time algorithms to solve the problem for graphs of degeneracy at most two and for the case where . These results imply linear-time algorithms for series-parallel graphs and grid graphs. In addition, we give linear-time algorithms for chordal graphs and cographs. On the other hand, we show that, for any , this problem remains NP-hard for planar graphs with degeneracy three and maximum degree four. Thus, we obtain a complexity dichotomy for this problem with respect to the degeneracy of a graph and the number k of colors.

Semi-proper orientations of dense graphs

An orientation D of a graph G is a digraph obtained from G by replacing each edge by exactly one of the two possible arcs with the same ends. An orientation D of a graph G is a k-orientation if the in-degree of each vertex in D is at most k. An orientation D of G is proper if any two adjacent vertices have different in-degrees in D. The proper orientation number of a graph G, denoted by →χ (G), is the minimum k such that G has a proper k-orientation.A weighted orientation of a graph G is a pair (D, w), where D is an orientation of G and w is an arc-weighting A(D) → N \\ {0}. A semi-proper orientation of G is a weighted orientation (D, w) of G such that for every two adjacent vertices u and v in G, we have that S(d,w)(v) ≠ S(d,w)(u), where S(d,w)(v) is the sum of the weights of the arcs in (D, w) with head v. For a positive integer k, a semi-proper k-orientation (D, w) of a graph G is a semi-proper orientation of G such that maxvϵV(G) S(d,w)(v) ≤ k. The semi-proper orientation number of a graph G, denoted by →χs(G), is the least k such that G has a semi-proper k-orientation.In this work, we first prove that →χs(G) ϵ {ω(G) - 1, ω(G)} for every split graph G, and that, given a split graph G, deciding whether →χs(G) = ω(G) - 1 is an NP-complete problem. We also show that, for every k, there exists a (chordal) graph G and a split subgraph H of G such that →χ(G) ≤ k and →χ(H) = 2k - 2. In the sequel, we show that, for every n ≥ p(p + 1), →χs(Ppn) = [3/2 p], where Ppn is the pth power of the path on n vertices. We investigate further unit interval graphs with no big clique: we show that →χ(G) ≤ 3 for any unit interval graph G with ω(G) = 3, and present a complete characterization of unit interval graphs with →χ(G)= ω(G) = 3. Then, we show that deciding whether →χs(G) = ω(G) can be solved in polynomial time in the class of co-bipartite graphs. Finally, we prove that computing →χs(G) is FPT when parameterized by the minimum size of a vertex cover in G or by the treewidth of G. We also prove that not only computing →χs(G) but also →χ(G), admits a polynomial kernel when parameterized by the neighbourhood diversity plus the value of the solution. These results imply kernels of size 40(k2) and 0(2kk2), in chordal graphs and split graphs, respectively, for the problem of deciding whether →χs(G) ≤ k parameterized by k. We also present exponential kernels for computing both →χ(G) and →χs(G) parameterized by the value of the solution when G is a cograph. On the other hand, we show that computing →χs(G) does not admit a polynomial kernel parameterized by the value of the solution when G is a chordal graph, unless NP ⊆ coNP/poly.

Axiomatic characterizations of Ptolemaic and chordal graphs

The interval function and the induced path function are two well studied class of set functions of a connected graph having interesting properties and applications to convexity, metric graph theory. Both these functions can be framed as special instances of a general set function termed as a transit function defined on the Cartesian product of a non-empty set \(V\) to the power set of \(V\) satisfying the expansive, symmetric and idempotent axioms. In this paper, we propose a set of independent first order betweenness axioms on an arbitrary transit function and provide characterization of the interval function of Ptolemaic graphs and the induced path function of chordal graphs in terms of an arbitrary transit function. This in turn gives new characterizations of the Ptolemaic and chordal graphs.

Minimal obstructions for a matrix partition problem in chordal graphs

In this article we consider graph partitions with prescribed patterns. Specifically we consider a 2-colouring of the vertex set of a graph such that one colour induces an independent set and the other colour induces a complete bipartite graph. We determine the complete list of obstructions (minimal forbidden induced subgraphs) for chordal graphs. This result completes the knowledge on minimal obstructions for 3 × 3 matrix partition problems on chordal graphs.

Locally identifying coloring in some chordal graphs

Locally identifying coloring in some chordal graphs

Some recent advances in random walks and random environments

Recent contributions to random walks in random environments and related topics are presented. We focus on non parametric estimation for one dimensional random walks in random environment and on the Dirichlet distribution on decomposable graphs.

On the computational difficulty of the terminal connection problem

A connection tree of a graph G for a terminal set W is a tree subgraph T of G such that leaves(T) ⊆ W ⊆ V(T). A non-terminal vertex is called linker if its degree in T is exactly 2, and it is called router if its degree in T is at least 3. The Terminal connection problem (TCP) asks whether G admits a connection tree for W with at most ℓ linkers and at most r routers, while the Steiner tree problem asks whether G admits a connection tree for W with at most k non-terminal vertices. We prove that, if r ≥ 1 is fixed, then TCP is polynomial-time solvable when restricted to split graphs. This result separates the complexity of TCP from the complexity of Steiner tree, which is known to be NP-complete on split graphs. Additionally, we prove that TCP is NP-complete on strongly chordal graphs, even if r ≥ 0 is fixed, whereas Steiner tree is known to be polynomial-time solvable. We also prove that, when parameterized by clique-width, TCP is W[1]-hard, whereas STeiner tree is known to be in FPT. On the other hand, agreeing with the complexity of Steiner tree, we prove that TCP is linear-time solvable when restricted to cographs (i.e. graphs of clique-width 2). Finally, we prove that, even if either ℓ ≥ 0 or r ≥ 0 is fixed, TCP remains NP-complete on graphs of maximum degree 3.

On general packing functions in graphs (Brief Announcement)

In this paper, we present the Generalized Packing Problem (GPF) in graphs in such a way that all other packing problems in graphs presented in the literature correspond to particular instances of it. We find the first class of graphs where one of these previously defined packing problems is tractable and another is NP-hard. We show that GPF remains linear-time solvable in strongly chordal graphs and, when reduced to instances with bounded neighborhood capacities (M-GPF), remains polynomial-time solvable in bounded clique-width graphs. To design specific algorithms that take advantage of the structural properties of certain subclasses of bounded clique-width graphs, we obtain technical results related to the behavior of the new packing parameter under several graph operations. In particular, we present a linear-time algorithm to solve GPF in block graphs and a polynomial-time algorithm to solve M-GPF in P4-tidy graphs.

A Remark on the Characterization of Triangulated Graphs

A Remark on the Characterization of Triangulated Graphs

О генерической сложности проблемы разбиения графа на треугольники

Generic-case approach to algorithmic problems was suggested by A. Miasnikov, V. Kapovich, P. Schupp, and V. Shpilrain in 2003. This approach studies behavior of an algorithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we study the generic complexity of the graph triangulation problem. This problem is as follows. Given a finite simple graph with 3n vertices, determine whether the vertices of the graph can be divided into n three-element sets, each of which contains vertices connected by edges of the original graph (that is, they are triangles). NP-completeness of this problem was proved by Shaffer in 1974 and is mentioned in the classic monograph by M. Garey and D. Johnson. We prove that under the conditions P ≠ NP and P = BPP there is no polynomial strongly generic algorithm for this problem. A strongly generic algorithm solves a problem not on the whole set of inputs, but on a subset whose frequency sequence converges exponentially to 1 with increasing size. To prove the theorem, we use the method of generic amplification, which allows one to construct generically hard problems from the problems that are hard in the classical sense. The main component of this method is the cloning technique, which combines the inputs of a problem together into sufficiently large sets of equivalent inputs. Equivalence is understood in the sense that the problem for them is solved in a similar way.

Weighted Modulo Orientations of Graphs and Signed Graphs

Given a graph $G$ and an odd prime $p$, for a mapping $f: E(G) \to {\mathbb Z}_p\setminus\{0\}$ and a ${\mathbb Z}_p$-boundary $b$ of $G$, an orientation $\tau$ is called an $(f,b;p)$-orientation if the net out $f$-flow is the same as $b(v)$ in ${\mathbb Z}_p$ at each vertex $v\in V(G)$ under orientation $D$. This concept was introduced by Esperet et al. (2018), generalizing mod $p$-orientations and closely related to Tutte's nowhere zero 3-flow conjecture. They proved that $(6p^2 - 14p + 8)$-edge-connected graphs have all possible $(f,b;p)$-orientations. In this paper, the framework of such orientations is extended to signed graph through additive bases. We also study the $(f,b;p)$-orientation problem for some (signed) graphs families including complete graphs, chordal graphs, series-parallel graphs and bipartite graphs, indicating that much lower edge-connectivity bound still guarantees the existence of such orientations for those graph families.

Intersecting longest paths in chordal graphs

We consider the size of the smallest set of vertices required to intersect every longest path in a chordal graph. Such sets are known as longest path transversals. We show that if ω(G) is the clique number of a chordal graph G, then there is a transversal of order at most 4⌈ω(G)5⌉. We also consider the analogous question for longest cycles, and show that if G is a 2-connected chordal graph then there is a transversal intersecting all longest cycles of order at most 2⌈ω(G)3⌉.

TreeMerge: A Visual Comparative Analysis Method for Food Classification Tree in Pesticide Residue Maximum Limit Standards

Food classification is an important part of food safety standards. In this paper, we propose a novel visual comparative analysis method for food classification trees (FCTs) in pesticide maximum residue limit (MRL) standards, called TreeMerge, to lay the foundation for a comprehensive comparison of pesticide MRL standards. First, a union tree is constructed by combining the two FCTs to be compared. Then, sunburst with an embedded chordal graph (SECG) and overlapping circular treemap (OCT), which are two new visualization solutions designed in this paper, are used to show the similarities and differences in a union tree. SECG can express the hierarchical structure and the similarity between corresponding nodes in the union tree at the same time. OCT uses an improved nested Venn diagram (overlapping circle) to express the attribute values in each layer of the union tree and uses a circle-filling layout algorithm based on the testing circle to improve the readability and space utilization of the view. Finally, a visual analysis system for comparing FCT, named FCTvis, is designed and implemented to support the exploration of the structural difference pattern of food classification in the two MRL standards and the quantity or scale of residue limits in various foods. The effectiveness of TreeMerge was verified by case studies on pesticide MRL standards in the Chinese Mainland and Chinese Hong Kong.

Fair Allocation of Indivisible Items with Conflict Graphs

We consider the fair allocation of indivisible items to several agents and add a graph theoretical perspective to this classical problem. Namely, we introduce an incompatibility relation between pairs of items described in terms of a conflict graph. Every subset of items assigned to one agent has to form an independent set in this graph. Thus, the allocation of items to the agents corresponds to a partial coloring of the conflict graph. Every agent has its own profit valuation for every item. Aiming at a fair allocation, our goal is the maximization of the lowest total profit of items allocated to any one of the agents. The resulting optimization problem contains, as special cases, both Partition and Independent Set. In our contribution we derive complexity and algorithmic results depending on the properties of the given graph. We show that the problem is strongly NP-hard for bipartite graphs and their line graphs, and solvable in pseudo-polynomial time for the classes of chordal graphs, cocomparability graphs, biconvex bipartite graphs, and graphs of bounded treewidth. Each of the pseudo-polynomial algorithms can also be turned into a fully polynomial approximation scheme (FPTAS).

Composition and product of cover-incomparability graphs

Cover-Incomparability graphs (C-I graphs) form an interesting class of graphs from posets. C-I graphs are identified among chordal graphs, distance-hereditary graphs, Ptolemaic graphs, split graphs, threshold graphs, bisplit graphs, block graphs and cographs. Thus only a few classes of graphs are known to be C-I graphs so far. Composition operation and various graph products are usually used to produce more non-trivial graphs in a particular graph class, using the prime graphs in the class. In this paper, we attempt to study the effect of the composition, lexicographic and strong products of C-I graphs. We found that the lexicographic product of two C-I graphs, say G and H is a C-I graph if and only if either G is any C-I graph, and H is a complete graph or vice versa. A similar result holds for the strong product also. It can be observed that the composition operation is more general than lexicographic product and we obtain new classes of C-I graphs from this operation.