46 research outputs found
Indicated domination game
Motivated by the success of domination games and by a variation of the
coloring game called the indicated coloring game, we introduce a version of
domination games called the indicated domination game. It is played on an
arbitrary graph by two players, Dominator and Staller, where Dominator
wants to finish the game in as few rounds as possible while Staller wants just
the opposite. In each round, Dominator indicates a vertex of that has
not been dominated by previous selections of Staller, which, by the rules of
the game, forces Staller to select a vertex in the closed neighborhood of .
The game is finished when all vertices of become dominated by the vertices
selected by Staller. Assuming that both players are playing optimally according
to their goals, the number of selected vertices during the game is the
indicated domination number, , of .
We prove several bounds on the indicated domination number expressed in terms
of other graph invariants. In particular, we find a place of the new graph
invariant in the well-known domination chain, by showing that for all graphs , and by showing that the indicated domination
number is incomparable with the game domination number and also with the upper
irredundance number. In connection with the trivial upper bound , we characterize the class of graphs attaining the bound
provided that . We prove that in trees, split graphs and
grids the indicated domination number equals the independence number. We also
find a formula for the indicated domination number of powers of paths, from
which we derive that there exist graphs in which the indicated domination
number is arbitrarily larger than the upper irredundance number.Comment: 19 page
Total dominator total coloring of a graph
Here, we initiate to study the total dominator total coloring of a graph which is a total coloring of the graph such that each object of the graph is adjacent or incident to every object of some color class. In more details, while in section 2 we present some tight lower and upper bounds for the total dominator total chromatic number of a graphs in terms of some parameters such as order, size, the total dominator chromatic and total domination numbers of the graph and its line graph, in section 3 we restrict our to trees and present a Nordhaus-Gaddum-like relation for trees. Finally in last section we show that there exist graphs that their total dominator total chromatic numbers are equal to their orders
Advances in Discrete Applied Mathematics and Graph Theory
The present reprint contains twelve papers published in the Special Issue “Advances in Discrete Applied Mathematics and Graph Theory, 2021” of the MDPI Mathematics journal, which cover a wide range of topics connected to the theory and applications of Graph Theory and Discrete Applied Mathematics. The focus of the majority of papers is on recent advances in graph theory and applications in chemical graph theory. In particular, the topics studied include bipartite and multipartite Ramsey numbers, graph coloring and chromatic numbers, several varieties of domination (Double Roman, Quasi-Total Roman, Total 3-Roman) and two graph indices of interest in chemical graph theory (Sombor index, generalized ABC index), as well as hyperspaces of graphs and local inclusive distance vertex irregular graphs
Future value based single assignment program representations and optimizations
An optimizing compiler internal representation fundamentally affects the clarity, efficiency and feasibility of optimization algorithms employed by the compiler. Static Single Assignment (SSA) as a state-of-the-art program representation has great advantages though still can be improved. This dissertation explores the domain of single assignment beyond SSA, and presents two novel program representations: Future Gated Single Assignment (FGSA) and Recursive Future Predicated Form (RFPF). Both FGSA and RFPF embed control flow and data flow information, enabling efficient traversal program information and thus leading to better and simpler optimizations. We introduce future value concept, the designing base of both FGSA and RFPF, which permits a consumer instruction to be encountered before the producer of its source operand(s) in a control flow setting. We show that FGSA is efficiently computable by using a series T1/T2/TR transformation, yielding an expected linear time algorithm for combining together the construction of the pruned single assignment form and live analysis for both reducible and irreducible graphs. As a result, the approach results in an average reduction of 7.7%, with a maximum of 67% in the number of gating functions compared to the pruned SSA form on the SPEC2000 benchmark suite. We present a solid and near optimal framework to perform inverse transformation from single assignment programs. We demonstrate the importance of unrestricted code motion and present RFPF. We develop algorithms which enable instruction movement in acyclic, as well as cyclic regions, and show the ease to perform optimizations such as Partial Redundancy Elimination on RFPF
Tight Bounds for MIS in Multichannel Radio Networks
Daum et al. [PODC'13] presented an algorithm that computes a maximal
independent set (MIS) within
rounds in an -node multichannel radio network with communication
channels. The paper uses a multichannel variant of the standard graph-based
radio network model without collision detection and it assumes that the network
graph is a polynomially bounded independence graph (BIG), a natural
combinatorial generalization of well-known geographic families. The upper bound
of that paper is known to be optimal up to a polyloglog factor.
In this paper, we adapt algorithm and analysis to improve the result in two
ways. Mainly, we get rid of the polyloglog factor in the runtime and we thus
obtain an asymptotically optimal multichannel radio network MIS algorithm. In
addition, our new analysis allows to generalize the class of graphs from those
with polynomially bounded local independence to graphs where the local
independence is bounded by an arbitrary function of the neighborhood radius.Comment: 37 pages, to be published in DISC 201