let g=(v,e) be a simple graph. an edge labeling f:e to {0,1} induces a vertex labeling f^+:v to z_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(1)-v_f(0) is called the edge-f...

We perform a systematic study in the computational complexity of the connected variant of three related transversal problems: Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. Just like their original counterparts, these variants are NP-complete for general graphs. A graph G is H-free for some graph H if G contains no induced subgraph isomorphic to H . It is known that Connected Ver...

Let G(V,E) be a graph with p vertices and q edges. A vertex labeling of G is an assignment f : V (G) → {1, 2, 3, . . . , p + q} be an injection. For a vertex labeling f, the induced Smarandachely edge m-labeling f S for an edge e = uv, an integer m ≥ 2 is defined by f ∗ S(e) = ⌈ f(u) + f(v) m ⌉ . Then f is called a Smarandachely super m-mean labeling if f(V (G))∪ {f(e) : e ∈ E(G)} = {1, 2, 3, ....

Common presentations of the NP-completeness of SAT suffer from two drawbacks which hinder the scope of this flagship result. First, they do not apply to machines equipped with random-access memory, also known as direct-access memory, even though this feature is critical in basic algorithms. Second, they incur a quadratic blow-up in parameters, even though the distinction between, say, linear an...

We present a comparative study of implementations of the following sorting algorithms on the Parsytec SC320 reconfigurable, asynchronous, massively parallel MIMD machine: Bitonic Sort, Odd-Even Merge Sort, Odd-Even Merge Sort with guarded split&merge, and two variants of Samplesort. The experiments are performed on 2up to 5-dimensional wrapped butterfly networks with 8 up to 160 processors. We ...

Let G be a graph with p vertices and q edges and let = 0,1, 2, , . 2 q A              A vertex labeling   : f V G A  i e a vertex equitable labeling of G if it induces an edge labeling s said to b given by      * =  f uv f u f v  * f such that     1 v a v b   and     * = 1,2,3, , f f f E q  , where   f v a is the ber of verti num ces v with   = f v a for . ...

We derive closed formulae for the numbers of rooted maps with a fixed number of vertices of the same odd degree except for the root vertex and one other exceptional vertex of degree 1. The same applies to the generating functions for these numbers. Similar results, but without the vertex of degree 1, were obtained by the first author and Rahman. We also show, by manipulating a recursion of Bout...

Let γ(G) be the domination number of a graph G. A graph G is domination-vertex-critical, or γ-vertex-critical, if γ(G− v) < γ(G) for every vertex v ∈ V (G). In this paper, we show that: Let G be a γ-vertex-critical graph and γ(G) = 3. (1) If G is of even order and K1,6-free, then G has a perfect matching; (2) If G is of odd order and K1,7-free, then G has a near perfect matching with only three...

A subset D of the vertex set V of a graph is called an open oddd dominating set if each vertex in V is adjacent to an odd number of vertices in D (adjacency is irreflexive). In this paper we solve the existence and enumeration problems for odd open dominating sets (and analogously defined even open dominating sets) in the m × n grid graph and prove some structural results for those that do exis...