Partition the vertices of a graph into one independent set and one acyclic set

For a given graph G, if the vertices of G can be partitioned into an independent set and an acyclic set, then we call G a near-bipartite graph. This paper studies the recognition of near-bipartite graphs. We give simple characterizations for those near-bipartite graphs having maximum degree at most 3 and those having diameter 2. We also show that the recognition of near-bipartite graphs is NP-complete even for graphs where the maximum degree is 4 or where the diameter is 4.     

Valid inequalities for a single constrained 0-1 MIP set intersected with a conflict graph

In this paper a mixed integer set resulting from the intersection of a single constrained mixed 0–1 set with the vertex packing set is investigated. This set arises as a subproblem of more general mixed integer problems such as inventory routing and facility location problems. Families of strong valid inequalities that take into account the structure of the simple mixed integer set and that of the vertex packing set simultaneously are introduced. In particular, the well-known mixed integer rounding inequality is generalized to the case where incompatibilities between binary variables are present. Exact and heuristic algorithms are designed to solve the separation problems associated to the proposed valid inequalities. Preliminary computational experiments show that these inequalities can be useful to reduce the integrality gaps and to solve integer programming problems.     

Problems on matchings and independent sets of a graph

Let $G$ be a finite simple graph. For $X?V\left(G\right)$, the difference of $X$, $d\left(X\right)?\left|X\right|?\left|N\left(X\right)\right|$ where $N\left(X\right)$ is the neighborhood of $X$ and $max\{d\left(X\right):X?V\left(G\right)\}$ is called the critical difference of $G$. $X$ is called a critical set if $d\left(X\right)$ equals the critical difference and $\ker \left(G\right)$ is the intersection of all critical sets. $\mathrm{diadem}\left(G\right)$ is the union of all critical independent sets. An independent set $S$ is an inclusion minimal set with$d\left(S\right)>0$ if no proper subset of $S$ has positive difference.A graph $G$ is called a König–Egerváry graph if the sum of its independence number $\alpha \left(G\right)$ and matching number $\mu \left(G\right)$ equals $\left|V\left(G\right)\right|$.In this paper, we prove a conjecture which states that for any graph the number of inclusion minimal independent set $S$ with $d\left(S\right)>0$ is at least the critical difference of the graph.We also give a new short proof of the inequality $\left|\ker \left(G\right)\right|+\left|\mathrm{diadem}\left(G\right)\right|\le 2\alpha \left(G\right)$.A characterization of unicyclic non-König–Egerváry graphs is also presented and a conjecture which states that for such a graph $G$, the critical difference equals $\alpha \left(G\right)?\mu \left(G\right)$, is proved.We also make an observation about $\ker \left(G\right)$ using Edmonds–Gallai Structure Theorem as a concluding remark.     

Generalizing D-graphs

Let ω₀(G) denote the number of odd components of a graph G. The deficiency of G is defined as def(G)=max_X⊆V(G)(ω₀(G-X)-|X|), and this equals the number of vertices unmatched by any maximum matching of G. A subset X⊆V(G) is called a Tutte set (or barrier set) of G if def(G)=ω₀(G-X)-|X|, and an extreme set if def(G-X)=def(G)+|X|. Recently a graph operator, called the D-graph D(G), was defined that has proven very useful in examining Tutte sets and extreme sets of graphs which contain a perfect matching. In this paper we give two natural and related generalizations of the D-graph operator to all simple graphs, both of which have analogues for many of the interesting and useful properties of the original.     

The stable set problem and the thinness of a graph

We introduce a poly-time algorithm for the maximum weighted stable set problem, when a certain representation is given for a graph. The algorithm generalizes the algorithm for interval graphs and that for graphs with bounded pathwidth. By a suitable application to the frequency assignment problem, we improved several solutions to relevant benchmark instances.     

Extensions of barrier sets to nonzero roots of the matching polynomial

In matching theory, barrier sets (also known as Tutte sets) have been studied extensively due to their connection to maximum matchings in a graph. For a root θ of the matching polynomial, we define θ-barrier and θ-extreme sets. We prove a generalized Berge-Tutte formula and give a characterization for the set of all θ-special vertices in a graph.     

The deficiency of a regular graph

A proper edge coloring c:E(G)→Z of a finite simple graph G is an interval coloring if the colors used at each vertex form a consecutive interval of integers. Many graphs do not have interval colorings, and the deficiency of a graph is an invariant that measures how close a graph comes to having an interval coloring. In this paper we search for tight upper bounds on the deficiencies of k-regular graphs in terms of the number of vertices. We find exact values for 1?k?4 and bounds for larger k.     

On the stable set polytope of a series-parallel graph

We give a short proof of Chvátal's conjecture that the nontrivial facets of the stable set polytope of a series-parallel graph all come from edges and odd holes.Research supported in part by the Natural Sciences and Engineering Research Council of Canada and by CP Rail.     

Measure preserving homomorphisms and independent sets in tensor graph powers

In this note, we study the behavior of independent sets of maximum probability measure in tensor graph powers. To do this, we introduce an upper bound using measure preserving homomorphisms. This work extends some previous results concerning independence ratios of tensor graph powers.     

Continuity at the extreme points

We prove that a bounded convex lower semicontinuous function defined on a convex compact set K is continuous at a dense subset of extreme points. If there is a bounded strictly convex lower semicontinuous function on K, then the set of extreme points contains a dense completely metrizable subset.     

The pinnacle set of a permutation

The peak set of a permutation records the indices of its peaks. These sets have been studied in a variety of contexts, including recent work by Billey, Burdzy, and Sagan, which enumerated permutations with prescribed peak sets. In this article, we look at a natural analogue of the peak set of a permutation, instead recording the values of the peaks. We define the “pinnacle set” of a permutation $w$ to be the set $\{w\left(i\right):i\text{ is a peak of}w\}$. Although peak sets and pinnacle sets mark the same phenomenon for a given permutation, the behaviors of these sets differ in notable ways as distributions over the symmetric group. In the work below, we characterize admissible pinnacle sets and study various enumerative questions related to these objects.     

The local dimensions of some Moran measures with open set condition

Dai and Liu obtained the formula of local dimensions of some Moran measures on Moran sets in R^d under the strong separation condition. In this paper, we prove that the result is still true under the open set condition. Due to the lack of the strong separation condition, our approach is essentially different to that used by Dai and Liu. We also obtain the formulas of the Hausdorff and packing dimensions of the Moran measures and discuss some interesting examples.     

Closed formulas for the numbers of small independent sets and matchings and an extremal problem for trees

We derive closed formulas for the numbers of independent sets of size at most 4 and matchings of size at most 3 in graphs without small cycles that depend only on the degree sequence and the products of the degrees of adjacent vertices.

As a related problem we describe an algorithm that determines a tree of given degree sequence that maximizes the sum of the products of the degrees of adjacent vertices.     

Rank of maximum matchings in a graph

The matching polytope is the convex hull of the incidence vectors of all (not necessarily perfect) matchings of a graphG. We consider here the problem of computing the dimension of the face of this polytope which contains the maximum cardinality matchings ofG and give a good characterization of this quantity, in terms of the cyclomatic number of the graph and families of odd subsets of the nodes which are always nearly perfectly matched by every maximum matching.This is equivalent to finding a maximum number of linearly independent representative vectors of maximum matchings ofG; the size of such a set is called thematching rank ofG. We also give in the last section a way of computing that rank independently of those parameters.Note that this gives us a good lower bound on the number of those matchings.     

On the exact Hausdorff measure of a class of self-similar sets satisfying open set condition

In this paper,we provide a new effective method for computing the exact value of Hausdorff measures of a class of self-similar sets satisfying the open set condition(OSC).As applications,we discuss a self-similar Cantor set satisfying OSC and give a simple method for computing its exact Hausdorff measure.     

A note on the computational complexity of graph vertex partition

A stable set of a graph is a vertex set in which any two vertices are not adjacent. It was proven in [A. Brandstädt, V.B. Le, T. Szymczak, The complexity of some problems related to graph 3-colorability, Discrete Appl. Math. 89 (1998) 59-73] that the following problem is NP-complete: Given a bipartite graph G, check whether G has a stable set S such thatG-Sis a tree. In this paper we prove the following problem is polynomially solvable: Given a graph G with maximum degree 3 and containing no vertices of degree 2, check whether G has a stable set S such thatG-Sis a tree. Thus we partly answer a question posed by the authors in the above paper. Moreover, we give some structural characterizations for a graph G with maximum degree 3 that has a stable set S such that G-S is a tree.     

Improved FPT algorithms for weighted independent set in bull-free graphs

Very recently, Thomassé et al. (2017) have given an FPT algorithm for Weighted Independent Set in bull-free graphs parameterized by the weight of the solution, running in time $2^{O\left(k^5\right)}?n^9$. In this article we improve this running time to $2^{O\left(k^2\right)}?n^7$. As a byproduct, we also improve the previous Turing-kernel for this problem from $O\left(k^5\right)$ to $O\left(k^2\right)$. Furthermore, for the subclass of bull-free graphs without holes of length at most $2p?1$ for $p\ge 3$, we speed up the running time to $2^{O(k?k^{\frac{1}{p?1}})}?n^7$. As $p$ grows, this running time is asymptotically tight in terms of $k$, since we prove that for each integer $p\ge 3$, Weighted Independent Set cannot be solved in time $2^{o\left(k\right)}?n^{O\left(1\right)}$ in the class of $\{\text{bull},C_4,\dots ,C_{2p?1}\}$-free graphs unless the ETH fails.