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For a simple, undirected, connected graph G , a function h : V (G) → { 0 , 1 , 2 , 3 } which satisfies the following conditions is called a total Roman {3}-dominating function (TR3DF) of G with weight h (V) = ∑ p ∈ V h (p) : (C1) For every vertex u ∈ V if h (u) = 0 , then u has m (m ≥ 1) neighbors such that whose sum is at least 3, and if h (u) = 1 , then u has n (n ≥ 1) neighbors such that whose sum is at least 2. (C2) The subgraph induced by the set of vertices labeled one, two or three has no isolated vertices. For a graph G , the smallest possible weight of a TR3DF of G denoted γ t { R 3 } (G) is known as the total Roman { 3 } -domination number of G. The problem of determining γ t { R 3 } (G) of a graph G is called minimum total Roman {3}-domination problem (MTR3DP). In this paper, we show that the problem of deciding if G has a TR3DF of weight at most l for chordal graphs is NP-complete. We also show that MTR3DP is polynomial time solvable for bounded treewidth graphs, chain graphs and threshold graphs. We design a 3 (ln (Δ − 0. 5) + 1. 5) -approximation algorithm for the MTR3DP and show that the same cannot have (1 − δ) ln | V | ratio approximation algorithm for any δ > 0 unless NP ⊆ DTIME (| V | O (log log | V |)). Next, we show that MTR3DP is APX-complete for graphs with Δ = 4. We also show that the domination and total Roman {3}-domination problems are not equivalent in computational complexity aspects. Finally, we present an integer linear programming formulation for MTR3DP. [ABSTRACT FROM AUTHOR]

Abstract: There are various techniques for reconstructing phylogenetic trees from data, and in this context the problem of determining how distant two such trees are from each other arises naturally. Various metrics for measuring the distance between two phylogenies have been defined. Another way of comparing two trees and is to compute the so called maximum agreement forest of these trees. Informally, the number of components of an agreement forest tells how many edges from each of and need to be cut so that the resulting forests agree, after performing all forced edge contractions. This problem is -hard even when the input trees have maximum degree 2. Hein et al. [J. Hein, T. Jiang, L. Wang, K. Zhang, On the complexity of comparing evolutionary trees, Discrete Applied Mathematics 71 (1996) 153–169] presented an approximation algorithm for it, claimed to have performance ratio 3. We show that the performance ratio of the algorithm proposed by Hein et al. is 4, and we also present two new 3-approximation algorithms for this problem. We show how to modify one of the algorithms into a -approximation algorithm for trees with bounded degree , . Finally, we report on some computational experiments comparing the performance of the algorithms presented in this paper. [Copyright &y& Elsevier]