A set P of edge-disjoint paths in a graph G is called an edge-covering of G if every edge of G is contained in a path of P. The path edge-covering number of G is then defined as the smallest number p(G) of paths in such an edge-covering of G. This parameter appeared in the study of certain models of information retrieval structures (see, e. g., [6]). If we are considering a tree T , then it is ...

Area minimization is one of the main efficiency criterion for lightweight encryption primitives. While reducing the implementation data path is a natural strategy for achieving this goal, Substitution-Permutation Network (SPN) ciphers are usually hard to implement in a bit-serial way (1-bit data path). More generally, this is hard for any data path smaller than its Sbox size, since many scan fl...

A path in an edge-colored graph G is rainbow if no two edges of the path are colored the same. The rainbow connection number rc(G) of G is the smallest integer k for which there exists a k-edge-coloring of G such that every pair of distinct vertices of G is connected by a rainbow path. Let f(d) denote the minimum number such that rc(G) ≤ f(d) for each bridgeless graph G with diameter d. In this...

A path in an edge colored graph is said to be a rainbow path if every edge in this colored with the same color. A vertex-colored graph G is rainbow vertex-connected if any pair of vertices in G are connected by a path whose internal vertices have distinct colors. The rainbow vertexconnection number of G denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainb...

We prove an exact lower bound on γ(G), the size of the smallest matching that a certain O(m+ n) time greedy matching procedure may find for a given graph G with n vertices and m edges. The bound is precisely Erdös and Gallai’s extremal function that gives the size of the smallest maximum matching, over all graphs with n vertices and m edges. Thus the greedy procedure is optimal in the sense tha...

We prove an exact lower bound on (G), the size of the smallest matching that a certain O(m + n) time greedy matching procedure may nd for a given graph G with n vertices and m edges. The bound is precisely Erdd os and Gallai's extremal function that gives the size of the smallest maximum matching, over all graphs with n vertices and m edges. Thus the greedy procedure is optimal in the sense tha...

We study the geometric shortest path and the minimum spanning tree problem with neighborhoods in the L1 metric. In this setting, we are given a graph G = (R, E), where R = {R1, . . . , Rn} is a set of polygonal regions in the plane. Placing a point pi inside each region Ri turns G into an edge-weighted graph Gp, p = {p1, . . . , pn}, where the cost of an edge is the distance between the points....

Let T be a planar subdivision with n vertices. Each face of T has a weight from [1, ρ] ∪ {∞}. A path inside a face has cost equal to the product of its length and the face weight. In general, the cost of a path is the sum of the subpath costs in the faces intersected by the path. For any ε ∈ (0, 1), we present a fully polynomial-time approximation scheme that finds a (1 + ε)-approximate shortes...

The forgotten topological index of a molecular graph $G$ is defined as $F(G)=sum_{vin V(G)}d^{3}(v)$, where $d(u)$ denotes the degree of vertex $u$ in $G$. The first through the sixth smallest forgotten indices among all trees, the first through the third smallest forgotten indices among all connected graph with cyclomatic number $gamma=1,2$, the first through<br /...

A graph with signed edges is orientation embedded in a surface when it is topologically embedded so that one trip around a closed path preserves or reverses orientation according as the path's sign product is positive or negative. We nd the smallest surface within which it is possible to orientation-embed the complete bipartite signed graph K r;s , which is obtained from the complete bipartite ...