The Behavior of Tree-Width and Path-Width Under Graph Operations and Graph Transformations

Graph Operations, Graph Transformations and Monadic Second-Order Logic A survey

Graph Operations, Graph Transformations and Monadic Second-Order Logic:: A survey

This dissertation focuses on integer flow problems within specific signed graphs. The theory of integer flows, which serves as a dual problem to vertex coloring of planar graphs, was initially introduced by Tutte as a tool related to the Four-Color Theorem. This theory has been extended to signed graphs. In 1983, Bouchet proposed a conjecture asserting that every flow-admissible signed graph admits a nowhere-zero 6-flow. To narrow dawn the focus, we investigate cubic signed graphs in Chapter 2. We prove that every flow-admissible 3-edge-colorable cubic signed graph admits a nowhere-zero 10-flow. This together with the 4-color theorem implies that every flow-admissible bridgeless planar signed graph admits a nowhere-zero 10-flow. As a byproduct of this research, we also demonstrate that every flow-admissible hamiltonian signed graph can admit a nowhere-zero 8-flow. In Chapter 3, we delve into triangularly connected signed graphs. Here, A triangle-path in a graph G is defined as a sequence of distinct triangles $T_1,T_2,\ldots,T_m$ in G such that for any i, j with $1\leq i < j \leq m$, $|E(T_i)\cap E(T_{i+1})|=1$ and $E(T_i)\cap E(T_j)=\emptyset$ if $j > i+1$. We categorize a connected graph $G$ as triangularly connected if it can be demonstrated that for any two nonparallel edges $e$ and $e'$, there exists a triangle-path $T_1T_2\cdots T_m$ such that $e\in E(T_1)$ and $e'\in E(T_m)$. For ordinary graphs, Fan {\it et al.} characterized all triangularly connected graphs that admit nowhere-zero $3$-flows or $4$-flows. Corollaries of this result extended to integer flow in certain families of ordinary graphs, such as locally connected graphs due to Lai and certain types of products of graphs due to Imrich et al. In this dissertation, we extend Fan's result for triangularly connected graphs to signed graphs. We proved that a flow-admissible triangularly connected signed graph $(G,\sigma)$ admits a nowhere-zero $4$-flow if and only if $(G,\sigma)$ is not the wheel $W_5$ associated with a specific signature. Moreover, this result is proven to be sharp since we identify infinitely many unbalanced triangularly connected signed graphs that can admit a nowhere-zero 4-flow but not 3-flow.\\ Chapter 4 investigates integer flow problems within $K_4$-minor free signed graphs. A minor of a graph $G$ refers to any graph that can be derived from $G$ through a series of vertex and edge deletions and edge contractions. A graph is considered $K_4$-minor free if $K_4$ is not a minor of $G$. While Bouchet's conjecture is known to be tight for some signed

Autonomous sequential networks are represented by transformation graphs, i.e., finite directed graphs, each vertex of which has out degree one. A network that can be realized by a set of simpler, parallel networks corresponds to a transformation graph representable as a cartesian product of simpler graphs. An algorithm for obtaining the various decompositions of a given transformation graph into irreducible factors is presented. The particular decompositions into a minimum number of factors and into factors of smallest size are directly reached by the method.

In this chapter, a very important class of graphs called intersection graph is introduced. Based on the geometrical representation, many different types of intersection graphs can be defined with interesting properties. Some of them—interval graphs, circular-arc graphs, permutation graphs, trapezoid graphs, chordal graphs, line graphs, disk graphs, string graphs—are presented here. A brief introduction of each of these intersection graphs along with some basic properties and algorithmic status are investigated.

From matchings to independent sets

In this paper, we introduce a new concept of dimension on fractal networks, the so-called Hausdorff Steiner [Formula: see text]-dimension. It is defined by a natural generalization of the Hausdorff dimension in Zeng’s work. In particular, this new dimension is a non-trivial higher estimate for the Hausdorff dimension. We explore its basic properties and determine the exact values of some networks. Meanwhile, we discuss the changes under different graph operations, including single node sum and the Cartesian product of graphs. Within a specific class of networks, the Hausdorff Steiner [Formula: see text]-dimension and the Hausdorff dimension exhibit an exact quantitative relationship.

We describe without proof polynomially bounded algorithms for the following problems: (i) (k is a fixed integer, andS is a fixed surface). With input a graph G which may be drawn in S and k pairs of vertices of G, decide if there are k vertex-disjoint paths of G, each joining one of the pairs of vertices; (ii) (His a fixed planar graph). With input a graph G, decide if G can be reduced to a graph isomorphic to H by deletion and contraction of edges.

For fixed integers r,ell geq 0, a graph G is called an (r,ell)-graph if the vertex set V(G) can be partitioned into r independent sets and ell cliques. This brings us to the following natural parameterized questions: Vertex (r,ell)-Partization and Edge (r,ell)-Partization. An input to these problems consist of a graph G and a positive integer k and the objective is to decide whether there exists a set S subseteq V(G) (S subseteq E(G)) such that the deletion of S from G results in an (r,ell)-graph. These problems generalize well studied problems such as Odd Cycle Transversal, Edge Odd Cycle Transversal, Split Vertex Deletion and Split Edge Deletion. We do not hope to get parameterized algorithms for either Vertex (r, ell)-Partization or Edge (r,ell)-Partization when either of r or ell is at least 3 as the recognition problem itself is NP-complete. This leaves the case of r,ell in {1,2}. We almost complete the parameterized complexity dichotomy for these problems by obtaining the following results: - We show that Vertex (r,ell)-Partization is fixed parameter tractable (FPT) for r,ell in {1,2}. Then we design an Oh(sqrt log n)-factor approximation algorithms for these problems. These approximation algorithms are then utilized to design polynomial sized randomized Turing kernels for these problems. - Edge (r,ell)-Partization is FPT when (r,ell)in{(1,2),(2,1)}. However, the parameterized complexity of Edge (2,2)-Partization remains open. For our approximation algorithms and thus for Turing kernels we use an interesting finite forbidden induced graph characterization, for a class of graphs known as (r,ell)-split graphs, properly containing the class of (r,ell)-graphs. This approach to obtain approximation algorithms could be of an independent interest.

Unit interval editing is fixed-parameter tractable

Finding large degree-anonymous subgraphs is hard

In this paper, we study the behaviour of the generalized power domination number of a graph by small changes on the graph, namely edge and vertex deletion and edge contraction. We prove optimal bounds for $\gamma_{p,k}(G-e)$, $\gamma_{p,k}(G/e)$ and for $\gamma_{p,k}(G-v)$ in terms of $\gamma_{p,k}(G)$, and give examples for which these bounds are tight. We characterize all graphs for which $\gamma_{p,k}(G-e) = \gamma_{p,k}(G)+1$ for any edge $e$. We also consider the behaviour of the propagation radius of graphs by similar modifications.

A graph with a vertex set is said to be a prime graph if there exists a bijective mapping , where denotes the number of vertices in , such that for any two adjacent vertices and in have . Tensor Product graph is a way to combine (compose) two graphs into one larger and more complex graph. The result is a new graph that reflects the connection properties of the two original graphs, but in a very specific and more complex way than other graph operations. Therefore, this research aims to determine whether there is prime labeling in the class of graphs resulting from the Tensor Product of the path graph and the cycle graph . The research employed analytical and exploratory methods with a trial-and-error strategy to determine the labeling that possesses a prime property. The results of this study prove that two classes of the Tensor Product graph for , and graph , for are prime graph. This finding expands the results on classes of graphs that admit prime labeling and provides a basis for further research on graph labeling in other graph operations

A vertex coloring c : V(G) → ℕ of a non-trivial connected graph G is called a set coloring if NC(u) ≠ NC(v) for any pair of adjacent vertices u and v. Here, NC(x) denotes the set of colors assigned to vertices adjacent to x. The set chromatic number of G, denoted by χs (G), is defined as the fewest number of colors needed to construct a set coloring of G. In this paper, we study the set chromatic number in relation to two graph operations: join and comb prdocut. We determine the set chromatic number of wheels and the join of a bipartite graph and a cycle, the join of two cycles, the join of a complete graph and a bipartite graph, and the join of two bipartite graphs. Moreover, we determine the set chromatic number of the comb product of a complete graph with paths, cycles, and large star graphs.

Distributed frameworks, such as MapReduce and Spark, have been developed by industry and research groups to analyze the vast amount of data that is being generated on a daily basis. Many graphs of interest, such as the Web graph and Social Networks, increase their size daily at an unprecedented scale and rate. To cope with this vast amount of data, researchers have been using distributed processing frameworks to analyze these graphs extensively. Most of these graph algorithms are iterative in nature. In our previous work, we introduced an efficient design pattern to handle a family of iterative graph algorithms in a distributed framework. Unfortunately, in most of these iterative algorithms, such as for Page-Rank, if the graph is modified with the addition or deletion of edges or vertices, the Page-Rank has to be recomputed from scratch. In this paper, we are introducing an improved design pattern for such algorithms to handle graph updates in an incremental fashion. Our method is to separate the graph topology from the graph analysis results. At each iteration step, each node participating in this graph analysis task, in addition to reading a single graph partition, it reads all the current analysis results from the distributed file system (DFS). These results are correlated with the local graph partition using a special merge-join and the new improved analysis results are calculated and stored to the DFS, one partition from each worker node. To handle continuous updates, an update function collects the changes to the graph and applies them to the graph partitions in a streaming fashion. Once the changes are made, the iterative algorithm is resumed to process the new updated data. Since a large part of the graph analysis task has already been completed on the existing data, the new updates require fewer iterations to compute the new graph analysis results as the iterative algorithm will converge faster.