Simplicial decompositions of graphs: a survey of applications

AbstractWe survey applications of simplicial decompositions (decompositions by separating complete subgraphs) to problems in graph theory. Among the areas of application are excluded minor theorems, extremal graph theorems, chordal and interval graphs, infinite graph theory and algorithmic aspects

Mobile vs. point guards

We study the problem of guarding orthogonal art galleries with horizontal mobile guards (alternatively, vertical) and point guards, using "rectangular vision". We prove a sharp bound on the minimum number of point guards required to cover the gallery in terms of the minimum number of vertical mobile guards and the minimum number of horizontal mobile guards required to cover the gallery. Furthermore, we show that the latter two numbers can be calculated in linear time.Comment: This version covers a previously missing case in both Phase 2 &

Chromatic number of the product of graphs, graph homomorphisms, Antichains and cofinal subsets of posets without AC

We have observations concerning the set theoretic strength of the following combinatorial statements without the axiom of choice. 1. If in a partially ordered set, all chains are finite and all antichains are countable, then the set is countable. 2. If in a partially ordered set, all chains are finite and all antichains have size $\aleph_{\alpha}$, then the set has size $\aleph_{\alpha}$ for any regular $\aleph_{\alpha}$. 3. CS (Every partially ordered set without a maximal element has two disjoint cofinal subsets). 4. CWF (Every partially ordered set has a cofinal well-founded subset). 5. DT (Dilworth's decomposition theorem for infinite p.o.sets of finite width). 6. If the chromatic number of a graph $G_{1}$ is finite (say $k<\omega$), and the chromatic number of another graph $G_{2}$ is infinite, then the chromatic number of $G_{1}\times G_{2}$ is $k$. 7. For an infinite graph $G=(V_{G}, E_{G})$ and a finite graph $H=(V_{H}, E_{H})$, if every finite subgraph of $G$ has a homomorphism into $H$, then so has $G$. Further we study a few statements restricted to linearly-ordered structures without the axiom of choice.Comment: Revised versio

Euclidean distance geometry and applications

Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of the most important applications: molecular conformation, localization of sensor networks and statics.Comment: 64 pages, 21 figure

Maximal independent sets, variants of chain/antichain principle and cofinal subsets without AC

In set theory without the Axiom of Choice (AC), we observe new relations of the following statements with weak choice principles. • Plf,c (Every locally finite connected graph has a maximal independent set). • Plc,c (Every locally countable connected graph has a maximal independent set). • CACאα (If in a partially ordered set all antichains are finite and all chains have size אα, then the set has size אα) if אα is regular. • CWF (Every partially ordered set has a cofinal well-founded subset). • If G = (VG, EG) is a connected locally finite chordal graph, then there is an ordering <of VG such that {w < v : {w, v} ∈ EG} is a clique for each v ∈ VG

The node-deletion problem for hereditary properties is NP-complete

AbstractWe consider the family of graph problems called node-deletion problems, defined as follows; For a fixed graph property Π, what is the minimum number of nodes which must be deleted from a given graph so that the resulting subgraph satisfies Π? We show that if Π is nontrivial and hereditary on induced subgraphs, then the node-deletion problem for Π is NP-complete for both undirected and directed graphs