On Independent Circuits Contained in a Graph

Agraph G is a diameter graph in ℝd if its vertex set is a finite subset in ℝd of diameter 1 and edges join pairs of vertices a unit distance apart. It is shown that if a diameter graph G in ℝ4 contains the complete subgraph K on five vertices, then any triangle in G shares a vertex with K. The geometric interpretation of this statement is as follows. Given any regular unit simplex on five vertices and any regular unit triangle in ℝ4, then either the simplex and the triangle have a common vertex or the diameter of the union of their vertex sets is strictly greater than 1.

Let G = (V(G),E(G)) denote a graph whose set of vertices and set of edges are V(G)and E(G), respectively. For any v ∈ V(G), we denote the neighbors of v as N(v) .B yn(G), we denote the number of vertices of G. All graphs considered here are both finite and simple. We denote, respectively, by Sn, Pn, and Cn the star, path, and cycle with n vertices. Let (G1,v1) and (G2,v2) be two graphs rooted at v1 and v2, respectively, then G = (G1,v1) �� (G1,v2) denote the graph obtained by identifying v1 with v2 as one common vertex. Let Un denote the set of all unicycle graphs of order n .B yU(n,k) we denote the set of unicycle graphs in which the length of its cycle is k. For any graph G in U(n,k), we denote the unique cycle of length k in G as Ck. Other notations and terminology not defined here will conform to those in [1]. For any given graph G, its Merrifield‐Simmons index, simply denoted as f(G), is defined to be the number of subsets of the vertex set, in which any two vertices are non-adjacent, i.e., in graph-theoretical terminology, the number of independent-vertex subsets of G, including the empty set. We take the cycle C4 =

Let G be a connected, undirected graph. Distance two labeling or a L(2,1)-labeling of a graph G is an assignment f from the vertex set V (G) to the set of non- negative integers such that |f(x)− f(y)| is greater than or equal to 2 if x and y are adjacent and |f(x) − f(y)| is greater than or equal to 1 if x and y are at distance 2, for all x and y in V (G). The L(2,1)-labeling number (Lamda)(G) of G is the smallest number k such that G has an L(2,1)-labeling f with max {f(v) : v 2 V (G) }is equal to k. In this paper, we present algorithms to get L(2,1)-labeling of cycle dominating graphs like Diamond graphs, nC4 with a common vertex and Books Bn and hence we find the (lamda)- number of these graphs. Keywords: Channel assignment; transmitters; L(2,1)-labeling; Books; Distance two labeling 2010 Mathematics Subject Classification: 05C78.

In a recent paper [l] K . CORRADI and A. HAJNAL proved that if a finite graph without multiple edges contains at least 3k vertices and the valency of every vertex is at least 2k, where k is a positive integer, then the graph contains k independent circuits, i . e . the graph contains as a subgraph a set of k circuits no two of which have a vertex in common . The present paper contains extensions of this theorem. In a recent paper [2] P . ERDŐS and L. POSA proved, among other things, that if a finite graph with or without loops and multiple edges contains n vertices and at least n + 4 edges, then the graph contains two circuits without an edge in common . The present paper contains analogous results for planar graphs . We adopt the following notation : O k denotes a graph consisting of k independent circuits, kO denotes a graph consisting of k or more circuits no two of which have an edge in common . If q is a graph then 'V (q) denotes the set of vertices of q, `V i(q ) denotes the set of vertices of q having valency i in q (i being a non-negative integer), T 5 i (q), Ty i (q) denote the set of vertices of 4 having valency i and ' i, respectively, and &(q) denotes the set of edges of c~ . The valency of the vertex x in the graph will be denoted by v (x, C~) . I ? (() I will be denoted by V (q), J & (q) I by E(q) etc. In this notation the theorem Of CORRADI and HAJNAL quoted above states that if q is a finite graph without multiple edges and if V(q) 3k and T -2k_ 1 (4) =0, then q Ok ; and the theorem of ERDOS and POSA quoted above states that if is a finite graph and E(C) _V(q) + 4, then qD 2 O .

Let ${\cal G}$ be a family of graphs defined on a common (labeled) vertex set $V$. A set $S\subset V$ is said to be a simultaneous strong metric generator for ${\cal G}$ if it is a strong metric generator for every graph of the family. The minimum cardinality among all simultaneous strong metric generators for ${\cal G}$, denoted by $Sd_s({\cal G})$, is called the simultaneous strong metric dimension of ${\cal G}$. We obtain general results on $Sd_s({\cal G})$ for arbitrary families of graphs, with special emphasis on the case of families composed by a graph and its complement. In particular, it is shown that the problem of finding the simultaneous strong metric dimension of families of graphs is $NP$-hard, even when restricted to families of trees.

The line graph L(G) of G has E(G) as its vertex set, and two vertices are adjacent in L(G) if and only if the corresponding edges share a common end vertex in G. Let σ¯2(G)=min{dG(x)+dG(y)|xy∈E(G)}. We show that, if σ¯2(G)≥2(⌊n11⌋−1) and n is sufficiently large, then either L(L(G)) is traceable or the Veldman’s reduction G′ is one of well-defined classes of exceptional graphs. Furthermore, if σ¯2(G)≥2(⌊n7⌋−1) and n is sufficiently large, then L(L(G)) is traceable. The bound 2(⌊n7⌋−1) is sharp. As a byproduct, we characterize the structure of a connected graph with a non-traceable 2-iterated line graph.

We initiate the study of degree-bounded network design problems in the online setting. The degree-bounded Steiner tree problem -- which asks for a subgraph with minimum degree that connects a given set of vertices -- is perhaps one of the most representative problems in this class. This paper deals with its well-studied generalization called the degree-bounded Steiner forest problem where the connectivity demands are represented by vertex pairs that need to be individually connected. In the classical online model, the input graph is given offline but the demand pairs arrive sequentially in online steps. The selected subgraph starts off as the empty subgraph, but has to be augmented to satisfy the new connectivity constraint in each online step. The goal is to be competitive against an adversary that knows the input in advance.The standard techniques for solving degree-bounded problems often fall in the category of iterative and dependent rounding techniques. Unfortunately, these rounding methods are inherently difficult to adapt to an online settings since the underlying fractional solution may change dramatically in between the rounding steps. Indeed, this might be the very reason that despite many advances in the online network design paradigm in the past two decades, the natural family of degree-bounded problems has remained widely open.In this paper, we design an intuitive greedy-like algorithm that achieves a competitive ratio of O(log n) where n is the number of vertices. We show that no (randomized) algorithm can achieve a (multiplicative) competitive ratio o(log n); thus our result is asymptotically tight. We further show strong hardness results for the group Steiner tree and the edge-weighted variants of degree-bounded connectivity problems.Furer and Raghavachari resolved the offline variant of degree-bounded Steiner forest in their paper in SODA'92. Since then, the family of degree-bounded network design problems has been extensively studied in the literature resulting in the development of many interesting tools and numerous papers on the topic. We hope that our approach and its dual analysis, paves the way for solving the online variants of the classical problems in this family of problems.

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A Roman k-dominating function on G is a function f : V (G) <TEX>$\rightarrow$</TEX> {0, 1, 2} such that every vertex u for which f(u) = 0 is adjacent to at least k vertices <TEX>$\upsilon_1,\;\upsilon_2,\;{\ldots},\;\upsilon_k$</TEX> with <TEX>$f(\upsilon_i)$</TEX> = 2 for i = 1, 2, <TEX>$\ldot$</TEX>, k. The weight of a Roman k-dominating function is the value f(V (G)) = <TEX>$\sum_{u{\in}v(G)}$</TEX> f(u). The minimum weight of a Roman k-dominating function on a graph G is called the Roman k-domination number <TEX>${\gamma}_{kR}$</TEX>(G) of G. Note that the Roman 1-domination number <TEX>$\gamma_{1R}$</TEX>(G) is the usual Roman domination number <TEX>$\gamma_R$</TEX>(G). In this paper, we investigate the properties of the Roman k-domination number. Some of our results extend these one given by Cockayne, Dreyer Jr., S. M. Hedetniemi, and S. T. Hedetniemi [2] in 2004 for the Roman domination number.

We want to find the vertex sets of components of a graph G with a known vertex set V and unknown edge set E. We learn about G by sending an oracle a query set $S \subseteq V$\hspace*{-1pt}, and the oracle tells us the vertices connected to S. The objective is to use the minimum number of queries to partition the vertex set into components. The problem is also known as interconnect diagnosis of wiring networks in VLSI. We present a deterministic algorithm using O(min{k, lg n}) queries and a randomized algorithm using expected O(min{k,lg k + lg lg n}) queries, where n is the number of vertices and k is the number of components. We also prove matching lower bounds.

Ordered Ramsey numbers

Our aim is to prove some results concerning the connectivity, lineconnectivity and J-connection or connection modulo J of total graphs. Only finite, nonoriented graphs without loops or multiple edges will be considered. Let G be a graph whose vertex set is V(G) and whose edge set is E(G). The elements of the set V(G)wE(G) will be called the elements of G and two elements of G are said to be associated if they are either adjacent or incident. The total graph T(G) of G is a graph whose vertex set is V(G)uE(G), two vertices being joined by an edge if and only if they are associated elements of G (see [1]). As an example of a graph G and its total graph T(G) see Fig. 1. We make an obvious distinction: small rings represent point-vertices of T(G) (x is a point-vertex x ~ E(G)). T(G) contains both G and its line-graph or interchange graph [4] L(G) as disjoint subgraphs. Remember that by definition, V[L(G)] =E(G) and two vertices of L(G) are linked by an edge if and only if the corresponding edges of G are adjacent. Edges of T(G) belonging neither to G nor L(G) form what will be called the incidence-graph I(G) of G. By definition the connectivity k(G) of G is the least number of vertices whose removal disconnects G or reduces G to a single vertex; a set of k(G) vertices satisfying this condition is called a minimal separating vertex set of G. Moreover G is n-connected if and only if k(G) >= n. On the

We address synthesis problems in the general context of network reliability, and describe two procedures − CENTERTREE and EXCENTER − whose running times never grow more than cubically with the number of vertices of the network. The first procedure solves two basic related problems − the most linking center and the most linking tree. The first problem requires locating a point on a network through which every connection between two given sets of vertices has to pass. This location maximizes the minimum reliability of the most reliable path connecting through the center every pair of vertices of the two given sets. The second problem asks for a most reliable tree connecting the two sets of vertices. The second procedure generalizes the first one and solves a constrained version of the most linking center problem that involves two (instead of one) reliabilities assigned to each network element and four (instead of two) sets of vertices to be connected. In general, the similarly extended, constrained version of the most linking tree problem is NP-hard, and only in a restricted case is it solvable by procedure EXCENTER. We prove that two extensions of the most linking tree problem, involving edge costs, are NP-hard.

We investigate a new class of codes for the optimal covering of vertices in an undirected graph G such that any vertex in G can be uniquely identified by examining the vertices that cover it. We define a ball of radius t centered on a vertex /spl upsi/ to be the set of vertices in G that are at distance at most t from /spl upsi/. The vertex /spl upsi/ is then said to cover itself and every other vertex in the ball with center /spl upsi/. Our formal problem statement is as follows: given an undirected graph G and an integer t/spl ges/1, find a (minimal) set C of vertices such that every vertex in G belongs to a unique set of balls of radius t centered at the vertices in C. The set of vertices thus obtained constitutes a code for vertex identification. We first develop topology-independent bounds on the size of C. We then develop methods for constructing C for several specific topologies such as binary cubes, nonbinary cubes, and trees. We also describe the identification of sets of vertices using covering codes that uniquely identify single vertices. We develop methods for constructing optimal topologies that yield identifying codes with a minimum number of codewords. Finally, we describe an application of the theory developed in this paper to fault diagnosis of multiprocessor systems.

A quasi-total Roman dominating function (QTRD-function) on is a function such that (i) every vertex x for which f(x) = 0 is adjacent to at least one vertex v for which f(v) = 2, and (ii) if x is an isolated vertex in the subgraph induced by the set of vertices with non-zero values, then f(x) = 1. The weight of a QTRD-function is the sum of its function values over the whole set of vertices, and the quasi-total Roman domination number is the minimum weight of a QTRD-function on G. The quasi-total Roman bondage number of a graph G is the minimum number of edges that have to be deleted to G in order to increase the quasi-total Roman domination number. In this paper, we start the study of quasi-total Roman bondage in graphs. We first show that the decision problem associated with the quasi-total Roman bondage problem is NP-hard even when restricted to bipartite graphs. Then basic properties of the quasi-total Roman bondage number are provided. Finally, some sharp bounds for are also presented.

On extremal cacti with respect to the revised Szeged index