Search results for "VERTEX"
showing 10 items of 225 documents
The Steiner Traveling Salesman Problem and its extensions
2019
Abstract This paper considers the Steiner Traveling Salesman Problem, an extension of the classical Traveling Salesman Problem on an incomplete graph where not all vertices have demand. Some extensions including several depots or location decisions are introduced, modeled and solved. A compact integer linear programming formulation is proposed for each problem, where the routes are represented with two-index decision variables, and parity conditions are modeled using cocircuit inequalities. Exact branch-and-cut algorithms are developed for all formulations. Computational results obtained confirm the good performance of the algorithms. Instances with up to 500 vertices are solved optimally.
Some properties of vertex-oblique graphs
2016
The type t G ( v ) of a vertex v ? V ( G ) is the ordered degree-sequence ( d 1 , ? , d d G ( v ) ) of the vertices adjacent with v , where d 1 ? ? ? d d G ( v ) . A graph G is called vertex-oblique if it contains no two vertices of the same type. In this paper we show that for reals a , b the class of vertex-oblique graphs G for which | E ( G ) | ? a | V ( G ) | + b holds is finite when a ? 1 and infinite when a ? 2 . Apart from one missing interval, it solves the following problem posed by Schreyer et?al. (2007): How many graphs of bounded average degree are vertex-oblique? Furthermore we obtain the tight upper bound on the independence and clique numbers of vertex-oblique graphs as a fun…
Chromatic Sums for Colorings Avoiding Monochromatic Subgraphs
2013
Abstract Given graphs G and H, a vertex coloring c : V ( G ) → N is an H-free coloring of G if no color class contains a subgraph isomorphic to H. The H-free chromatic number of G, χ ( H , G ) , is the minimum number of colors in an H-free coloring of G. The H-free chromatic sum of G , Σ ( H , G ) , is the minimum value achieved by summing the vertex colors of each H-free coloring of G. We provide a general bound for Σ ( H , G ) , discuss the computational complexity of finding this parameter for different choices of H, and prove an exact formulas for some graphs G. For every integer k and for every graph H, we construct families of graphs, G k with the property that k more colors than χ ( …
On the family ofr-regular graphs with Grundy numberr+1
2014
Abstract The Grundy number of a graph G , denoted by Γ ( G ) , is the largest k such that there exists a partition of V ( G ) , into k independent sets V 1 , … , V k and every vertex of V i is adjacent to at least one vertex in V j , for every j i . The objects which are studied in this article are families of r -regular graphs such that Γ ( G ) = r + 1 . Using the notion of independent module, a characterization of this family is given for r = 3 . Moreover, we determine classes of graphs in this family, in particular, the class of r -regular graphs without induced C 4 , for r ≤ 4 . Furthermore, our propositions imply results on the partial Grundy number.
Padding and the expressive power of existential second-order logics
1998
Padding techniques are well-known from Computational Complexity Theory. Here, an analogous concept is considered in the context of existential second-order logics. Informally, a graph H is a padded version of a graph G, if H consists of an isomorphic copy of G and some isolated vertices. A set A of graphs is called weakly expressible by a formula ϕ in the presence of padding, if ϕ is able to distinguish between (sufficiently) padded versions of graphs from A and padded versions of graphs that are not in A.
Decidability of bisimulation equivalences for parallel timer processes
1993
In this paper an abstract model of parallel timer processes (PTPs), allowing specification of temporal quantitative constraints on the behaviour of real time systems, is introduced. The parallel timer processes are defined in a dense time domain and are able to model both concurrent (with delay intervals overlapping on the time axis) and infinite behaviour. Both the strong and weak (abstracted from internal actions) bisimulation equivalence problems for PTPs are proved decidable. It is proved also that, if one provides the PTP model additionally with memory cells for moving timer value information along the time axis, the bisimulation equivalence (and even the vertex reachability) problems …
Refined Finiteness and Degree Properties in Graphs
2020
Summary In this article the finiteness of graphs is refined and the minimal and maximal degree of graphs are formalized in the Mizar system [3], based on the formalization of graphs in [4].
On Packing Colorings of Distance Graphs
2014
International audience; The {\em packing chromatic number} $\chi_{\rho}(G)$ of a graph $G$ is the least integer $k$ for which there exists a mapping $f$ from $V(G)$ to $\{1,2,\ldots ,k\}$ such that any two vertices of color $i$ are at distance at least $i+1$. This paper studies the packing chromatic number of infinite distance graphs $G(\mathbb{Z},D)$, i.e. graphs with the set $\mathbb{Z}$ of integers as vertex set, with two distinct vertices $i,j\in \mathbb{Z}$ being adjacent if and only if $|i-j|\in D$. We present lower and upper bounds for $\chi_{\rho}(G(\mathbb{Z},D))$, showing that for finite $D$, the packing chromatic number is finite. Our main result concerns distance graphs with $D=…
Polyhedral results for a vehicle routing problem
1991
Abstract The Vehicle Routing Problem is a well known, and hard, combinatorial problem, whose polyhedral structure has deserved little attention. In this paper we consider the particular case in which all the demands are equal (since in the general case the associated polytope may be empty). From a known formulation of the problem we obtain the dimension of the corresponding polytope and we study the facetial properties of every inequality in it.
On the hardness of optimization in power-law graphs
2008
Our motivation for this work is the remarkable discovery that many large-scale real-world graphs ranging from Internet and World Wide Web to social and biological networks appear to exhibit a power-law distribution: the number of nodes y"i of a given degree i is proportional to i^-^@b where @b>0 is a constant that depends on the application domain. There is practical evidence that combinatorial optimization in power-law graphs is easier than in general graphs, prompting the basic theoretical question: Is combinatorial optimization in power-law graphs easy? Does the answer depend on the power-law exponent @b? Our main result is the proof that many classical NP-hard graph-theoretic optimizati…