Note on the upper bound of the rainbow index of a graph

A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is a rainbow path if every two edges of it receive distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the minimum number of colors that are needed to color the edges of $G$ such that there exists a rainbow path connecting every two vertices of $G$. Similarly, a tree in $G$ is a rainbow~tree if no two edges of it receive the same color. The minimum number of colors that are needed in an edge-coloring of $G$ such that there is a rainbow tree connecting $S$ for each $k$-subset $S$ of $V(G)$ is called the $k$-rainbow index of $G$, denoted by $rx_k(G)$, where $k$ is an integer such that $2\leq k\leq n$. Chakraborty et al. got the following result: For every $\epsilon> 0$, a connected graph with minimum degree at least $\epsilon n$ has bounded rainbow connection, where the bound depends only on $\epsilon$. Krivelevich and Yuster proved that if $G$ has $n$ vertices and the minimum degree $\delta(G)$ then $rc(G)<20n/\delta(G)$. This bound was later improved to $3n/(\delta(G)+1)+3$ by Chandran et al. Since $rc(G)=rx_2(G)$, a natural problem arises: for a general $k$ determining the true behavior of $rx_k(G)$ as a function of the minimum degree $\delta(G)$. In this paper, we give upper bounds of $rx_k(G)$ in terms of the minimum degree $\delta(G)$ in different ways, namely, via Szemer\'{e}di's Regularity Lemma, connected $2$-step dominating sets, connected $(k-1)$-dominating sets and $k$-dominating sets of $G$.Comment: 12 pages. arXiv admin note: text overlap with arXiv:0902.1255 by other author

On Edge Dominating Number of Tensor Product of Cycle and Path

A subset S' of E(G) is called an edge dominating set ofG if every edge not in S' is adjacent to some edge in S'. The edge dominatingnumber of G, denoted by γ'(G), of G is the minimum cardinality takenover all edge dominating sets of G. Let G1 (V1, E1) and G2(V2,E2) betwo connected graph. The tensor product of G1 and G2, denoted byG1⨂▒G2 is a graph with the cardinality of vertex |V| = |V1| × |V2|and two vertices (u1,u2) and (v1,v2) in V are adjacent in G1⨂▒G2ifu1 v1 ∈ E1 and u2,v2 ∈E2 . In this paper we study an edge dominatingnumber in the tensor product of path and cycle. The results show thatγ'(Cn⨂▒P2) = ⌈2n/3⌉ for n is odd, γ'(Cn⨂▒P3) = n for n is odd, and theedge dominating number is undefined if n is even. For n ∈even number,we investigated the edge dominating number of its component on tensorproduct of cycle Cn and path. The results are γ'c(Cn⨂▒P2)= ⌈n/3⌉ andγ'c(Cn ⨂▒P3) = ⌈n/2⌉ which Cn ,P2 and P3, respectively, is Cycle order n,Path order 2 and Path order 3

In the complement of a dominating set

A set D of vertices of a graph G=(V,E) is a dominating set, if every vertex of D\V has at least one neighbor that belongs to D. The disjoint domination number of a graph G is the minimum cardinality of two disjoint dominating sets of G. We prove upper bounds for the disjoint domination number for graphs of minimum degree at least 2, for graphs of large minimum degree and for cubic graphs.A set T of vertices of a graph G=(V,E) is a total dominating set, if every vertex of G has at least one neighbor that belongs to T. We characterize graphs of minimum degree 2 without induced 5-cycles and graphs of minimum degree at least 3 that have a dominating set, a total dominating set, and a non-empty vertex set that are disjoint.A set I of vertices of a graph G=(V,E) is an independent set, if all vertices in I are not adjacent in G. We give a constructive characterization of trees that have a maximum independent set and a minimum dominating set that are disjoint and we show that the corresponding decision problem is NP-hard for general graphs. Additionally, we prove several structural and hardness results concerning pairs of disjoint sets in graphs which are dominating, independent, or both. Furthermore, we prove lower bounds for the maximum cardinality of an independent set of graphs with specifed odd girth and small average degree.A connected graph G has spanning tree congestion at most s, if G has a spanning tree T such that for every edge e of T the edge cut defined in G by the vertex sets of the two components of T-e contains at most s edges. We prove that every connected graph of order n has spanning tree congestion at most n^(3/2) and we show that the corresponding decision problem is NP-hard

A New Perspective on Vertex Connectivity

Edge connectivity and vertex connectivity are two fundamental concepts in graph theory. Although by now there is a good understanding of the structure of graphs based on their edge connectivity, our knowledge in the case of vertex connectivity is much more limited. An essential tool in capturing edge connectivity are edge-disjoint spanning trees. The famous results of Tutte and Nash-Williams show that a graph with edge connectivity $\lambda$ contains \floor{\lambda/2} edge-disjoint spanning trees. We present connected dominating set (CDS) partition and packing as tools that are analogous to edge-disjoint spanning trees and that help us to better grasp the structure of graphs based on their vertex connectivity. The objective of the CDS partition problem is to partition the nodes of a graph into as many connected dominating sets as possible. The CDS packing problem is the corresponding fractional relaxation, where CDSs are allowed to overlap as long as this is compensated by assigning appropriate weights. CDS partition and CDS packing can be viewed as the counterparts of the well-studied edge-disjoint spanning trees, focusing on vertex disjointedness rather than edge disjointness. We constructively show that every $k$-vertex-connected graph with $n$ nodes has a CDS packing of size $\Omega(k/\log n)$ and a CDS partition of size $\Omega(k/\log^5 n)$. We prove that the $\Omega(k/\log n)$ CDS packing bound is existentially optimal. Using CDS packing, we show that if vertices of a $k$-vertex-connected graph are independently sampled with probability $p$, then the graph induced by the sampled vertices has vertex connectivity $\tilde{\Omega}(kp^2)$. Moreover, using our $\Omega(k/\log n)$ CDS packing, we get a store-and-forward broadcast algorithm with optimal throughput in the networking model where in each round, each node can send one bounded-size message to all its neighbors

Dynamic Parameterized Problems

In this work, we study the parameterized complexity of various classical graph-theoretic problems in the dynamic framework where the input graph is being updated by a sequence of edge additions and deletions. Vertex subset problems on graphs typically deal with finding a subset of vertices having certain properties that are of interest to us. In real-world applications, the graph under consideration often changes over time and due to this dynamics, the solution at hand might lose the desired properties. The goal in the area of dynamic graph algorithms is to efficiently maintain a solution under these changes. Recomputing a new solution on the new graph is an expensive task especially when the number of modifications made to the graph is significantly smaller than the size of the graph. In the context of parameterized algorithms, two natural parameters are the size k of the symmetric difference of the edge sets of the two graphs (on n vertices) and the size r of the symmetric difference of the two solutions. We study the Dynamic Pi-Deletion problem which is the dynamic variant of the Pi-Deletion problem and show NP-hardness, fixed-parameter tractability and kernelization results. For specific cases of Dynamic Pi-Deletion such as Dynamic Vertex Cover and Dynamic Feedback Vertex Set, we describe improved FPT algorithms and give linear kernels. Specifically, we show that Dynamic Vertex Cover admits algorithms with running times 1.1740^k*n^{O(1)} (polynomial space) and 1.1277^k*n^{O(1)} (exponential space). Then, we show that Dynamic Feedback Vertex Set admits a randomized algorithm with 1.6667^k*n^{O(1)} running time. Finally, we consider Dynamic Connected Vertex Cover, Dynamic Dominating Set and Dynamic Connected Dominating Set and describe algorithms with 2^k*n^{O(1)} running time improving over the known running time bounds for these problems. Additionally, for Dynamic Dominating Set and Dynamic Connected Dominating Set, we show that this is the optimal running time (up to polynomial factors) assuming the Set Cover Conjecture

The kk-Dominating Graph

Given a graph $G$, the $k$-dominating graph of $G$, $D_k(G)$, is defined to be the graph whose vertices correspond to the dominating sets of $G$ that have cardinality at most $k$. Two vertices in $D_k(G)$ are adjacent if and only if the corresponding dominating sets of $G$ differ by either adding or deleting a single vertex. The graph $D_k(G)$ aids in studying the reconfiguration problem for dominating sets. In particular, one dominating set can be reconfigured to another by a sequence of single vertex additions and deletions, such that the intermediate set of vertices at each step is a dominating set if and only if they are in the same connected component of $D_k(G)$. In this paper we give conditions that ensure $D_k(G)$ is connected.Comment: 2 figure, The final publication is available at http://link.springer.co