How can we detect communities in graphs with edge-labels, such as time-evolving networks or edge-colored graphs? Unlike classical graphs, edge-labels contain additional information about the type of edges, e.g., when two people got connected, or which company hosts the air route between two cities. We model community detection on edge-labeled graphs as a tensor decomposition problem and propose...

We consider the Tree Augmentation problem: given a graph G = (V,E) with edge-costs and a tree T on V disjoint to E, find a minimum-cost edge-subset F ⊆ E such that T ∪ F is 2-edge-connected. Tree Augmentation is equivalent to the problem of finding a minimumcost edge-cover F ⊆ E of a laminar set-family. The best known approximation ratio for Tree Augmentation is 2, even for trees of radius 2. A...

‎‎let $g=(v,e)$ be a simple graph‎. ‎an edge labeling $f:eto {0,1}$ induces a vertex labeling $f^+:vtoz_2$ defined by $f^+(v)equiv sumlimits_{uvin e} f(uv)pmod{2}$ for each $v in v$‎, ‎where $z_2={0,1}$ is the additive group of order 2‎. ‎for $iin{0,1}$‎, ‎let‎ ‎$e_f(i)=|f^{-1}(i)|$ and $v_f(i)=|(f^+)^{-1}(i)|$‎. ‎a labeling $f$ is called edge-friendly if‎ ‎$|e_f(1)-e_f(0)|le 1$‎. ‎$i_f(g)=v_f(...

In 1990, Thomas proved that every graph admits a tree decomposition of minimum width additionally satisfies certain vertex-connectivity condition called leanness. This result had many uses and has been extended to several other decompositions. this paper, we consider tree-cut decompositions, have introduced by Wollan (2015) as possible edge-version We show an edge-connectivity analogous Thomas'

Let C(n;S) be a fixed circulant. The main problem is to determine the spectrum for values of v such that the complete graph Kv admits an edge-disjoint decomposition into subgraphs each of which is isomorphic to C(n;S). A complete solution in not to be expected since some instances correspond to famous existence problems (for instance, the existence of BIBD’s with λ = 1). On the other hand, the ...

We investigate the properties of chordal graphs that follow from the well-known fact that chordal graphs admit tree representations. In particular, we study the structure of reduced clique graphs which are graphs that canonically capture all tree representations of chordal graphs. We propose a novel decomposition of reduced clique graphs based on two operations: edge contraction and removal of ...

In this work, we study conditions for the existence of length-constrained path-cycle decompositions, that is, partitions of the edge set of a graph into paths and cycles of a given minimum length. Our main contribution is the characterization of the class of all triangle-free graphs with odd distance at least 3 that admit a path-cycle decomposition with elements of length at least 4. As a conse...

‎let $g=(v(g),e(g))$ be a simple connected graph with vertex set $v(g)$ and edge‎ ‎set $e(g)$‎. ‎the (first) edge-hyper wiener index of the graph $g$ is defined as‎: ‎$$ww_{e}(g)=sum_{{f,g}subseteq e(g)}(d_{e}(f,g|g)+d_{e}^{2}(f,g|g))=frac{1}{2}sum_{fin e(g)}(d_{e}(f|g)+d^{2}_{e}(f|g)),$$‎ ‎where $d_{e}(f,g|g)$ denotes the distance between the edges $f=xy$ and $g=uv$ in $e(g)$ and $d_{e}(f|g)=s...

Definition 1 A tree decomposition of a graph G(V,E) is a pair (T, χ) where T = (I, F ) is a tree and χ = {χi|i ∈ I} is a family of subsets of V such that (a) ⋃ i∈I χi = V , (b) for each edge e = {u, v} ∈ E, there exists i ∈ I such that both u and v belong to χi, and (c) for all v ∈ V , there is a set of nodes {i ∈ I|v ∈ χi} that forms a connected subtree of T . The width of a tree decomposition...

Throughout the paper, G denotes a graph that has no loops, but that may have multiple edges. The multi-set of edges of G is denoted by E(G). A graph G is simple if it has no (loops or) multiple edges. A biclique of G is a simple complete bipartite subgraph of G. A biclique decomposition of G is a collection of bicliques of G, such that each edge of G is in precisely one of the bicliques in the ...