(Total) Irredundant Set And Dominating Set In Graph

Domination number γ(G) , independant domination number i{G) , the (upper) total irredunance number (IR_t(G))ir_t(G) and the (upper) irredunadance number (IR(G))ir(G)a.re all important graphic structural parameters and has been studied for a long history.D.P.Sumner and P.Blitich conjectured in [10] that if G is a 3 - γ-critical graph ,then γ(G) = i(G).But this conjecture hasn't been proved till now.In [19] , Wang chunxiang et al gave a sufficient condition on which the above conjecture holds and raised a new conjecture that if G is a 3 - (γ, d)- critical graph ,then γ(G) = i(G) . In the first part of my master dissertation , I gave a new sufficient condition on which the first conjecture holds and a sufficient condition on which the second one holds when d = 2.In [30],S.M.Hedetniemi et al showed that it's a NP-hard problem to determine the (upper)total irredundance number for any given graph . In 2002 Odile Favaron et al studied the total irredundant set in theory. They've characterized the graph satisfying the equality ir_t{G) = IR_t(G) = 0 and the tree with ir_t(G) - 1. They also investigated the regular graph satisfying ir_t(G) > 1 at the same time. In the end, they questioned whether the bound of the (up-per)total irredundance number can be phrased in terms of the minimum degree S(G) ?In the second one of my paper , I mainly dealt with this question by giving two upper bounds for the (upper)total irredundance number in terms ofthe minimum degree 8(G) ,I showed that these two bounds are reachable and gave the necessary condition on which the bounds are attainable .In the third part of my paper , I investigated the stability number SN(G) of the upper irredundance IR(G)-the maximum edges E whose removal will result in IR{G - E') = IR(G) . I showed that(l)For any nonempty connected graph G with order n greater than or equal to 2 ,SN(G) ≤ n - 2.(2)SN(G) ≤ (IR{G) - l)△(G) - 1 holds if IR{G) ≥ 2.