Small independent edge dominating sets in graphs of maximum degree three

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1 Small independent edge dominating sets in graphs of maimum degree three Grażna Zwoźniak Institute of Computer Science Wrocław Universit Small independent edge dominating sets in graphs of maimum degree three p. 1/34

2 Problem definition Given a connected undirected graph G = (V, E) Find a minimum set of edges E E(G) which fulfils the following conditions: for ever edge (u, w) E(G) either (u, w) E or one of (, u), (w, ) E(G) belongs to E no two edges of E share a common endpoint. Small independent edge dominating sets in graphs of maimum degree three p. 2/34

3 Arbitrar graphs results Minimum independent edge dominating set problem is NP-complete even when restricted to planar or bipartite graphs of maimum degree three M. Yannakakis, F. Gavril, Journal on Appl. Math The problem remains NP-complete for planar bipartite graphs, line graphs, total graphs, planar cubic graphs J. D. Horton, K. Kilakos, Journal on Disc. Math There is a polnomial time algorithm for trees M. Yannakakis, F. Gavril, Journal on Appl. Math Small independent edge dominating sets in graphs of maimum degree three p. 3/34

4 Arbitrar graphs approimation It is NP-hard to approimate the problem within an factor smaller than 7/6 in general graphs and smaller than 1+1/487 in cubic graphs M. Chlebík, J. Chlebíková, Algorithms and Computations 2003 Constructing an maimal matching gives approimation ratio of 2 for the problem Small independent edge dominating sets in graphs of maimum degree three p. 4/34

5 Cubic graphs results The minimum independent edge dominating set of an n-verte cubic graph is at most 9n/20+O(1) W. Duckworth, N. C. Wormald, 2000 There are families of cubic graphs for which the size of the minimum independent edge dominating set is at least 3n/8 W. Duckworth, N. C. Wormald, 2000 The lower bound for the size of the minimum independent edge dominating set is 3n/10 Small independent edge dominating sets in graphs of maimum degree three p. 5/34

6 Our results An approimation algorithm which finds the independent edge dominating set of size at most 4n/9+1/3 in an n-verte graph of maimum degree three. As a collorar we obtain an approimation ratio of 40/27 for the problem in cubic graphs. Small independent edge dominating sets in graphs of maimum degree three p. 6/34

7 Rules (a) (b) (c) (d) (e) (f) T i T i T i T i T i T i v v v v v v u w u u u 1 u 1 1 u 1 w 1 w 2 w w u s u 2 w u s w Small independent edge dominating sets in graphs of maimum degree three p. 7/34

8 Construction of the forest F 1. F 2. V (T 0 ) {r 0, v 1, v 2, v 3 }, where r 0 is an degree three verte of G and v 1, v 2, v 3 Γ G (r 0 ); E(T 0 ) {(r 0, v 1 ), (r 0, v 2 ), (r 0, v 3 )}; let r 0 be a root of T 0 ; i 0 3. if it is possible find the leftmost leaf in T i that can be epanded b the Rule 1 and epand it; go to the step 3; else go to the step 4; 4. if it is possible find the leftmost leaf in T i that can be epanded b the Rule 2 and epand it; go to the step 3 else F F T i and go to the step 5 5. if it is possible find the leftmost leaf v in T i such that Rule 3 can be applied to v and appl this rule to v; i i + 1; let T i be a new tree created in this step; go to the step 3 with T i else go to the step 5 with the father of T i Small independent edge dominating sets in graphs of maimum degree three p. 8/34

9 Properties of F Fact 1 Let v V (T i ), T i F. 1. If deg Ti (v) = 2 and w Γ Ti (v) then deg Ti (w) = If deg Ti (v) = 2 and w Γ G (v) then w V (T i ). 3. If v is adjacent to the verte w EX R, u Γ G (v) and u w then u V (T i ). Small independent edge dominating sets in graphs of maimum degree three p. 9/34

10 The edges e E(G)\E(F) (a) (b) (c) (d) (e) (f) T i T i T i T i T j T j T j T j T j T j (g) (h) (i) (j) T i T i T i T i T j T j T j T j Small independent edge dominating sets in graphs of maimum degree three p. 10/34

11 Notation G current graph F current forest EX = V (G )\V (F ) S the subtree of T F rooted at some verte V (T ) EDS(G) independent edge dominating set contructed b our algorithm Small independent edge dominating sets in graphs of maimum degree three p. 11/34

12 DOMINATE i DOMINATE 1 (S S ) dominates the edges of the subtrees S, S, where deg T () = 2 and is the endpoint of the edge (, ) E(G)\E(F) together with edges e E(G )\E(F ) incident to V (S ) V (S ). DOMINATE 2 (S ) which dominates the edges of some paths which end at the leaves of S and go through the edges e E(G )\E(F ). DOMINATE 3 (S ) dominates E(S ) and all the edges incident to V (S ). Small independent edge dominating sets in graphs of maimum degree three p. 12/34

13 Main propert Propert 1 The procedure DOMINATE i (), i {1, 2, 3} adds m edges to EDS(G) and removes n vertices from V (G ), where m 4 9 n. The onl eception ma be the case when DOMINATE 3 () is applied to the tree rooted at r 0 or at the son of r 0 ; then m 4 9 n Small independent edge dominating sets in graphs of maimum degree three p. 13/34

14 Required edges (a) (b) (c) w w z u u z w V (F) u v L(S ) v EX R v Small independent edge dominating sets in graphs of maimum degree three p. 14/34

15 DOMINATE 3 (S ) - basic (a) (b) (c) (d) (e) (f) (g) Small independent edge dominating sets in graphs of maimum degree three p. 15/34

16 DOMINATE 3 (S ) when h(s ) = 2 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r) Small independent edge dominating sets in graphs of maimum degree three p. 16/34

17 S definition Let S be a subgraph of S which fulfils the following condition: if two edges (u 1, u 2 ), (v 1, v 2 ) E(S ) where either the path from u 2 to in T is shorter than the path from v 2 to in T or u 2 is on the left of v 2 in S, are incident to an edge (u 2, v 2 ) REQUIRED(S ), then the edge (u 1, u 2 ) E(S ). Small independent edge dominating sets in graphs of maimum degree three p. 17/34

18 DOMINATE 3 (S ) when h(s ) = 2 and = r (a) (b) (c) r r r u 1 u 1 u 2 w 1 w 1 w 2 Small independent edge dominating sets in graphs of maimum degree three p. 18/34

19 DOMINATE 3 (S ) when h(s ) = 2 (a) (b) (c) z 2 z 1 w z 2 z 1 z 2 z 1 w w z 1 w Small independent edge dominating sets in graphs of maimum degree three p. 19/34

20 Impossible to appl DOMINATE 3 (S ) (a) (b) (c) (d) (e) (f) z z z Small independent edge dominating sets in graphs of maimum degree three p. 20/34

21 DOMINATE 3 (S ) when h(s ) = 3 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) Small independent edge dominating sets in graphs of maimum degree three p. 21/34

22 DOMINATE 3 (S ) when h(s ) = 3 (a) (b) (c) (d) (e) w w w w w Small independent edge dominating sets in graphs of maimum degree three p. 22/34

23 DOMINATE(S ) when = r 0 (a) (b) (c) (d) (e) r 0 r 0 r 0 r 0 r 0 (f) (g) (h) (i) r 0 r 0 r 0 r 0 r 0 (j) (k) (l) r 0 r 0 Small independent edge dominating sets in graphs of maimum degree three p. 23/34

24 DOMINATE(S ) when is the last son of r 0 (a) (b) (c) (d) (e) (f) (g) (h) (i) Small independent edge dominating sets in graphs of maimum degree three p. 24/34

25 DOMINATE 1 (S S ) (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) Small independent edge dominating sets in graphs of maimum degree three p. 25/34

26 DOMINATE 2 (S ) (a) (b) (c) (d) S S S S v 4 v 6 u 1 w 1 u 1 w 1 w 2 v 3 v 2 v 1 v 5 v 4 v 3 v 2 v 1 v 1 v 2 v 3 v 1 v 2 v 3 v 4 v 5 (e) (f) (g) T T T w 2 S u 2 S u 2 S w 1 w 1 w 2 u 1 u 2 w 1 w 1 u 1 u 1 u 2 u 1 w 1 v 3 v 2 v 1 v 1 v 2 v 3 v 1 v 2 v 3 Small independent edge dominating sets in graphs of maimum degree three p. 26/34

27 EDS_T(T ) 1. appl procedure EDS_T() to the sons T i F of T (successivel from the leftmost to the rightmost); 2. EDS(T ). Small independent edge dominating sets in graphs of maimum degree three p. 27/34

28 EDS(S ) Let W = {u V (S ) : deg F (u) = 2 and deg G (u) = 3}. 1. while W (a) appl in-order search and find the first verte u W ; let w be such a verte that (u, w) E(G)\E(F); (b) EDS(S u); (c) if w V (F ): EDS(S w); (d) if deg G (u) = 3 or deg G (w) = 3: MARK_REQUIRED(S u S w); DOMINATE_PATHS(S u S w); DOMINATE 1 (S u S w); 2. MARK(S ). Small independent edge dominating sets in graphs of maimum degree three p. 28/34

29 MARK_REQUIRED(S ) Let W = {(u, w) E(G )\E(F ) : u L(S )}. 1. REQUIRED(S ) := 2. while there is a verte u L(S ) and the edges (u, v), (u, w) W, v L(S ), w L(S ): find the first u, add (u, w) to REQUIRED(S ) and remove the edges incident to u, w from W. 3. while there is a verte u L(S ) and the edge (u, w) W, w EX R: find the first u, add (u, w) to REQUIRED(S ) and remove the edges incident to u, w from W. 4. while there is a verte u L(S ) and the edge (u, w) W : find the first u, add (u, w) to REQUIRED(S ) and remove the edges incident to u, w from W. Small independent edge dominating sets in graphs of maimum degree three p. 29/34

30 DOMINATE_PATHS(S ) 1. if TAILS 1 (S ) or EARS 1 (S ) : DOMINATE 2 (S ); 2. MARK_REQUIRED(S ); 3. if TAILS 2 (S ) or EARS 2 (S ) or EARS 3 (S ) : DOMINATE 2 (S ); goto 2. Small independent edge dominating sets in graphs of maimum degree three p. 30/34

31 MARK(S ) 1. if h(s ) > 2: appl procedure MARK() to the trees rooted at the sons of (successivel from the leftmost to the rightmost); 2. MARK_REQUIRED(S ); 3. DOMINATE_PATHS(S ); 4. if has at most two sons u i, 1 i 2: (a) while DOMINATION_POSSIBLE(S u i ): DOMINATE 3 (S u i ); (b) if DOMINATION_POSSIBLE(S ): DOMINATE 3 (S ); else: return; 5. if has three sons u i, 1 i 3 (in this case = r 0 ): (a) while DOMINATION_POSSIBLE(S u i ): DOMINATE 3 (S u i ); (b) if h(s r 0 ) > 1 and deg G (r 0 ) = 3: let u 3 V ( S r 0 ); i. DOMINATE 3 ( S r 0 ); ii. if u 3 V (G ): MARK_REQUIRED(S u 3 ); DOMINATE 3 (S u 3 ); (c) else if r 0 V (G ): DOMINATE 3 (S rsmall 0 ). independent edge dominating sets in graphs of maimum degree three p. 31/34

32 Properties of F Fact 2 Let deg T (u) = deg T (w) = 2, T F, (u, w) E(G)\E(F), a = LCA T (u, w) and let u be on the left of w in T. Then w is the right son of a, and there are no vertices of degree 2 on the path from u to a in T. Small independent edge dominating sets in graphs of maimum degree three p. 32/34

33 Properties of F Fact 3 Let r be a root of a tree T F. Let u L(T) and let w be the first verte of degree 2 on the path from u to r in T. If there is v V (T) such that deg F (v) = 2 and (u, v) E(G)\E(T) then w is an ancestor of v in T. Small independent edge dominating sets in graphs of maimum degree three p. 33/34

34 Main theorem Theorem 1 Let G be a graph of maimum degree three, where V (G) = n. Our algorithm constructs for G an independent edge dominating set EDS(G) of size at most 4n/9 +1/3 in linear time. Small independent edge dominating sets in graphs of maimum degree three p. 34/34

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