on the algorithmic complexity of double vertex-edge ... · the concept of double vertex-edge...

On the Algorithmic Complexity of Double vertex-edgedomination in graphs

NARESH KUMAR HResearch Scholar

Research Supervisor : Dr.Y.B.VenkatakrishnanDepartment of Mathematics, SASH

Shanmugha Arts, Science, Technology and Research Academy (SASTRA)Thanjavur-613 401.

13th International Conference and Workshop on Algorithms and ComputationIndian Institute of Technology, Guwahati

February 27- March 02, 2019

Naresh Kumar (SASTRA) Algorithm - Double vertex-edge domination March 01, 2019 1 / 20

Definition

Vertex-edge domination:A vertex v vertex-edge dominates every edge uv incident to v, as well asevery edge adjacent to these incident edges.

Definition

A set S ⊆ V is a vertex-edge dominating set if for every edge e ∈ E,there exists a vertex v ∈ S such that v ve-dominates e. The minimumcardinality of a ve-dominating set of G, denoted by γve(G), is called thevertex-edge domination number of G.

The concept was introduced by Peters [13] in 1986 and was studied byLewis [9] in 2007. For, recent developments refer [1, 7].

Definition

Double vertex-edge domination

The concept of double vertex-edge domination was introduced byKrishnakumari et al, [8].

Definition

A set S ⊆ V is a double vertex-edge dominating set, abbreviated DVEDS,if every edge e ∈ E is ve-dominated by at least two vertices of S.

The double vertex-edge domination number of G, denoted byγdve(G), is the minimum cardinality of a double ve-dominating set ofG.

Figure: Example

Definition

Figure: Example

Definition

Figure: Example

Definition

Figure: Example

Definition

Figure: Example

Problems associated with double vertex edge domination

Minimum double vertex-edge domination problemInstance : A graph G = (V,E).Solution : A double vertex-edge dominating set of G, a subset V ′ ⊂ Vsuch that each edge e ∈ E(G) gets ve-dominated by at least two verticesof V ′.Measure : Cardinality of double vertex-edge dominating set, |V ′|.

Double Vertex-edge Domination Decision problem (DVEDD)Instance: A graph G = (V,E) and a positive integer k ≤ |V |.Question: Does there exist a DVED-set D in G such that |D| ≤ k?

Complexity Status

The DVEDD problem is NP-complete for bipartite graphs -Krishnakumari et al, in [8].

We now show that the DVEDD problem is NP-complete for chordalgraphs.

We use the idea Exact cover by 3-sets.

Exact cover by 3-sets (X3C)Instance: A finite set X with |X| = 3q and a collection C of 3-elementsubsets of X.Question: Does C contain an exact cover for X, that is, a sub-collectionC ′ ⊆ C such that for every element in X belongs to exactly one memberof C ′?

Theorem (Garey [2])

The X3C problem is NP-complete

Complexity Status

Theorem (Garey [2])

Complexity Status

Theorem (Garey [2])

Complexity Status

Theorem (Garey [2])

Complexity Status

Theorem (Garey [2])

Reduction from X3C to DVEDD

Theorem

The DVEDD problem is NP-complete for chordal graphs.

Given a set D, it can be verified in polynomial time whether D is adouble vertex-edge dominating set of G and |D| ≤ k.

Let X = {x1, x2, . . . , x3q} and C = {C1, C2, . . . , Ct} be an arbitraryinstance of X3C.

Let V (G) = {xi, yi, zi, pi, ti : 1 ≤ i ≤ 3q} ∪ {ci : 1 ≤ i ≤ t},E(G) = {xicj : xi ∈ Cj , 1 ≤ i ≤ 3q, 1 ≤ j ≤ t}∪{xiyi, yizi, zipi, piti :1 ≤ i ≤ 3q} ∪ {cicj : 1 ≤ i < j ≤ t; j = 1, 2, . . . , t} and k = 7q.

α = {t1, t2, . . . , t3q, p1, p2, . . . , p3q, z1, z2, . . . , z3q, y1, y2, . . . , y3q,x1, x2, . . . , x3q, c1, c2, . . . , cm} is a perfect elimination ordering of G

Theorem

The construction of the chordal graph G = (V,E) associated with theinstance of X3C, where X = {x1, x2, x3, x4, x5, x6} andC ={C1 = {x1, x2, x4}, C2 = {x2, x3, x6}, C3 = {x3, x4, x5}, C4 = {x3, x5, x6}}.

Figure: Double vertex-edge domination for chordal graphs

Algorithm for Proper Interval graphs

Let G be a connected proper interval graph with a BCOσ = (v1, v2, . . . , vn).

Two arrays D and S for selecting the vertices in DV E.

D[v] = 0 - at least one edge incident with v is not ve-dominated.

D[v] = 1 - at least one edge incident with v is ve-dominated once.

D[v] = 2 - all the edges incident with v are ve-dominated twice.

S[v] = 0 - then the vertex v is not in the set DV E, otherwiseS[v] = 1.

DVED-PROPER INTERVAL GRAPHS I

DV E = ∅;for i = 1 to n− 1 do

if D[vi] 6= 2 thenLet NGi(vi) = {vi1 , vi2 , . . . , vir−1 , vir}, where i1 < i2 < . . . < ir−1 < ir;if D[vi] = 0 then

if |NGi+1(vi+1)| ≥ 2 then

DV E = DV E ∪ {vi+1r−1, vi+1r};

D[x] = 2 for all x ∈ N [vi+1r−1] ∩N [vi+1r ];

D[x] = 1 for allx ∈

(N [vi+1r−1 ] ∪N [vi+1r ]

)\(N [vi+1r−1 ] ∩N [vi+1r ]

S[vi+1r−1] = S[vi+1r ] = 1;

end ifif |NGi+1

(vi+1)| = 1 thenDV E = DV E ∪ {vi+1, vi+2};D[x] = 2 for all x ∈ N [vi+1] ∩N [vi+2];D[x] = 1 for all x ∈ (N [vi+1] ∪N [vi+2]) \ (N [vi+1] ∩N [vi+2]);S[vi+1] = S[vi+1] = 1;

DVED-PROPER INTERVAL GRAPHS II

end ifif |NGi+1

(vi+1)| = 0 thenDV E = DV E ∪ {vi, vi+1};D[x] = 2 for all x ∈ NGi

[vi];S[vi] = S[vi+1] = 1;

end ifend ifif D[vi] = 1 then

if |NGi+1(vi+1)| = 0 then

if S[vi+1] = 1 thenDV E = DV E ∪ {vi}, D0[x] = 1 for all x ∈ N [vi];D1[x] = 2 for all x ∈ N [vi], S[vi] = 1;

elseDV E = DV E ∪ {vi+1}, D0[x] = 1 for all x ∈ N [vi+1];D1[x] = 2 for all x ∈ N [vi+1], S[vi+1] = 1;

end ifelse

if S[vi+1r ] = 1 thenDV E = DV E ∪ {vi+1r−1

}, D0[x] = 1 for all x ∈ N [vi+1r−1];

DVED-PROPER INTERVAL GRAPHS III

D1[x] = 2 for all x ∈ N [vi+1r−1], S[vi+1r−1

] = 1;elseDV E = DV E ∪ {vi+1r}, D0[x] = 1 for all x ∈ N [vi+1r ];D1[x] = 2 for all x ∈ N [vi+1r ], S[vi+1r ] = 1;

end ifend if

end ifend ifdelete vi, Gi ← Gi \ {vi};

end forreturn DV E.

The BCO σ = (v1, v2, . . . , vn) of a proper interval graph can becomputed in Linear time [11].

Each iteration of the algorithm DVED-PROPER INTERVALGRAPHS checks the degree of the vertex vi+1 in the graph Gi+1.

Thus the total time taken is O(n+m).

Approximation Hardness

Minimum vertex-edge domination problemInstance : A graph G = (V,E).Solution : A vertex-edge dominating set of G, a subset V ′ ⊂ V such thateach edge e ∈ E(G) gets ve-dominated by the vertices of V ′ .Measure : Cardinality of vertex-edge dominating set |V ′|.

Theorem (Lewis [9])

For a graph G = (V,E), the minimum vertex-edge domination problemcannot be approximated within (1− ε) ln |V | for any ε > 0 unless NP ⊆DTIME (|V |O(log log |V |)).

Approximation Hardness

Minimum vertex-edge domination problemInstance : A graph G = (V,E).Solution : A vertex-edge dominating set of G, a subset V ′ ⊂ V such thateach edge e ∈ E(G) gets ve-dominated by the vertices of V ′ .Measure : Cardinality of vertex-edge dominating set |V ′|.

Theorem (Lewis [9])

For a graph G = (V,E), the minimum vertex-edge domination problemcannot be approximated within (1− ε) ln |V | for any ε > 0 unless NP ⊆DTIME (|V |O(log log |V |)).

Figure: An example of a graph transformation for Approximation hardness

Theorem

For a graph G = (V,E), the minimum double vertex-edge dominationproblem cannot be approximated within (1− ε) ln |V | for any ε > 0 unlessNP ⊆ DTIME (|V |O(log log |V |)).

Figure: An example of a graph transformation for Approximation hardness

Theorem

For a graph G = (V,E), the minimum double vertex-edge dominationproblem cannot be approximated within (1− ε) ln |V | for any ε > 0 unlessNP ⊆ DTIME (|V |O(log log |V |)).

APX-Completeness

MIN DD SET-BInstance : A graph G = (V,E) with degree at most B.Solution : A double dominating set of G, a subset V ′ ⊂ V such that eachvertex in V (G) is dominated by at least two vertices of V ′.Measure : Cardinality of double dominating set, |V ′|.

Theorem (Klasing [6])

MIN DD SET-4 is APX-complete.

The corona of G with K1 transforms the minimum double dominating setof G to the double vertex dominating set of G ◦K1.

Theorem

MIN DVED SET-5 is APX-complete.

APX-Completeness

Theorem

APX-Completeness

Theorem

APX-Completeness

Theorem

Reference I

[1] Bourtig, R., Chellali, M., Haynes, T.W., Hedetniemi, S.:Vertex-Edge Domination in Graphs, Aequationes Mathematicae 90,355–366 (2016)

[2] Garey, M.R., Johnson, D.S., Computers and Interactability : A guideto the theory of NP -completeness, Freeman, New York, (1979)

[3] Haynes, T., Hedetniemi, S., Slater, P.: Fundamentals of Dominationin Graphs, Marcel Dekker, New York (1998)

[4] Haynes, T., Hedetniemi, S., Slater P.(eds.): Domination in Graphs:Advanced Topics, Marcel Dekker, New York, (1998)

[5] Jamison, R.E., Laskar, R.: Elimination orderings of chordal graphs,In: Proceedings of the seminar on Combinatorics and Applications,Indian Statistical Institute, Calcutta 192–200 (1982)

Reference II

[6] Klasing, R., Laforest, C.: Hardness results and approximationalgorithms of k-tuple domination in graphs, Information ProcessingLetters 89, 75-83 (2004)

[7] Krishnakumari, B., Chellali, M., Venkatakrishnan, Y.B.: DoubleVertex-Edge Domination, Discrete Mathematics, Algorithms andApplictions (2017). DOI: 10.1142/S1793830917500458.

[8] Krishnakumari, B., Venkatakrishnan,Y.B., Krzywkowski, M.: Boundson the vertex-edge domination number of a tree, Comptes RendusMathematique 352, 363–366 (2014)

[9] Lewis, J.R.: Vertex-edge and edge-vertex parameters in graphs,Ph.D. Thesis, Clemson University (2007)

[10] Lewis, J., Hedetniemi, S., Haynes, T., Fricke, G.: Vertex-edgedomination, Utilitas Mathematica 81, 193–213 (2010)

Reference III

[11] Panda, B.S., Das, S.K.: A linear time recognition algorithm forproper interval graphs, Inform. Process. Lett. 87(3), 153161 (2003)

[12] Papadimitriou, C.M., Yannakakis, M.: Optimization, approximationand complexity classes, J Comput System Sci 43, 425–440 (1991)

[13] Peters, J., Theoretical and Algorithmic Results on Domination andConnectivity, Ph.D. Thesis, Clemson University (1986)

Acknowledgment

The authors thank National Board for Higher Mathematics, Mumbaifor the support.

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