- Email: [email protected]
On an extension of distance-hereditary graphs
Electronic Notes in Discrete Mathematics 22 (2005) 253–259 www.elsevier.com/locate/endm
On an extension of distance-hereditary graphs Kahina Meslem 1 Facult´e des Math´ematiques U.S.T.H.B., B.P. 32 El Alia Algiers, Algeria
M´eziane A¨ıder 2 Facult´e des Math´ematiques U.S.T.H.B., B.P. 32 El Alia Algiers, Algeria
Abstract Given a simple and finite connected graph G, the distance dG (u, v) is the length of the shortest induced {u,v}-path linking the vertices u and v in G. Bandelt and Mulder [2] have characterized the class of distance hereditary graphs where the distance is preserved in each connected subgraph. In this paper, we are interested with the class of k-distance hereditary graphs (k ∈ N∗ ) which consists in a parametric extension of the distance-heredity notion. We allow the distance in each connected induced subgraph to increase by at most (k − 1) unities. We provide a characterization of k-distance hereditary graphs in terms of forbidden configurations for each k ∈ N∗ . Keywords: Distance heredity, Dilation number, Forbidden configurations.
1 2
Email: kahina [email protected] Email: [email protected]
1571-0653/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2005.06.056
254
1
K. Meslem, M. Aïder / Electronic Notes in Discrete Mathematics 22 (2005) 253–259
Introduction
The major problem that we meet in several networks consists of the inefficiency of transmitting messages between processors. Even if the sender and the destination are still connected, the messages are delivered with a certain delay. The distance is the main function which expresses the connectivity between the sites and the faulty tolerance in these networks. It is computed by means of shortest paths in networks. The k-distance hereditary graphs can be used to represent some networks communication. In fact, a network modeled as a k-distance hereditary graph (k ∈ N∗ ) is characterized as follows : if two sites have failed in transmission, as they remain connected, the distance between them in the faulty network is bounded by the distance in the original network plus an integer k −1, k ∈ N∗ . This means that a graph G belongs to k-distance hereditary class if and only if the length of each induced {u, v}-path is at most the distance in the original graph plus k − 1. Among this class, we can find the well known distance-hereditary graphs (k = 1) which are introduced by Howorka [5] and characterized in several ways by Bandelt and Mulder [2] using both algorithmic and metric aspect. Moreover, A¨ıder [1] has introduced and studied the 2-distance hereditary graphs (he called almost-distance hereditary graphs). Using the characterization of these graphs, other both combinatorial and metric proprieties were provided. Cicerone and Di Stefano [4] have used the notion of twin graph to characterize the k-distance hereditary graphs and have stated the coNP-completeness of the recognizing problem in this class. In this paper, we characterize k-distance hereditary graphs in terms of forbidden subgraphs using the chord distance in cycles and the dilation number of a graph. We provide all forbidden configurations for each integer k ≥ 1.
2
Preliminaries definitions
We only consider finite, simple loopless and undirected graphs G=(V ,E) where V is the vertex set and E is the edge set. A subgraph of G is a graph having all its vertices and edges in G. The neighborhood of a vertex u denoted by N (u) consists in all the vertices v which are adjacent to u. Given a subset S of V , the induced subgraph < S > of G is the maximal subgraph of G with vertex set S. An induced path of G is a non redundant connection of a pair of vertices. The length of a shortest path between two vertices u and v is called distance and is denoted by dG (u, v) ; moreover the length of a longest induced path between the same vertices is denoted by DG (u, v).
K. Meslem, M. Aïder / Electronic Notes in Discrete Mathematics 22 (2005) 253–259
255
A cycle Cn is a path (x1 , x2 , ..., xn , x1 ). For convenience, the (x, y) part of a path P is denoted by P (x.y). A chord of a cycle is an edge that joins non consecutive vertices of Cn . The chord distance of a cycle denoted by cd(Cn ) is defined to be the minimum number of consecutive vertices in Cn such that every chord of Cn is incident to some of such vertices. However, each cycle on n vertices with chord distance equal either to 0 or to 1 will be referred to as a 1Cn -configuration (see Fig. 1, where dashed lines represent eventual chords in the cycle). .. .. ... .. .. .. ... . . .. .. ... .. .. .. ... . . . .. . ... .. .. ... . ... .. ... ... .. .. .. ...... ...
Fig. 1. 1C5 -configurations
A pair {u, v} of non adjacent vertices is called cycle-pair if there exists a shortest induced {u, v}-path P and a longest induced {u, v}-path Q such that P ∩ Q = {u, v}. So, P ∪ Q induce a cycle denoted C{u, v}. The split composition of the graphs G1 = (V1 ∪ m1 , E1 ) and G2 = (V2 ∪ m2 , E2 ) with respect to m1 and m2 is the graph G1 ∗ G2 having vertex set V = V1 ∪ V2 and edge set E = E1 ∪ E2 ∪ {(u, v)/u ∈ N (m1 ); v ∈ N (m2 )}. (See Fig. 2).
m1
m2
Fig. 2. The split composition of two C5 with respect to m1 and m2
Definition 2.1 Let k be a positive integer. A graph G is k-distance hereditary if and only if for any connected induced subgraph H of G we have: ∀u, v ∈ H, dH (u, v) ≤ dG (u, v) + (k − 1) A¨ıder [1] has given a characterization of 2-distance hereditary graphs by providing all forbidden configurations, which are the 1Cn configuration for each
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K. Meslem, M. Aïder / Electronic Notes in Discrete Mathematics 22 (2005) 253–259
n ≥ 6 and the 2C5 -configurations. In Fig. 3 (and in all our figures) a dashed line represents an eventual edge and a dotted line represents a path, eventually empty (in this case, the end vertices coincide). Other characterizations in terms of forbidden subgraphs involving chord distance proprieties were established for each k-distance hereditary graphs (k ≥ 1). For instance, Cicerone and Di Stefano [4] showed that a graph G is kdistance hereditary if and only if the chord distance of each cycle Cn , with (n > k + 5), is more than n−k+1
− 1 in the twin graph G∗ . The twin graph 2 ∗ G is obtained by duplicating every vertex (by a false twin) of G. The implicit form of this result incites us to check for the forbidden subgraphs of k-distance hereditary graphs. That is the main target of the following section.
.. .. ... .. .. .. ... . . . .. . ... .. .. .. ... . .. .... . ... .. ... ... . .. ... ... .. .. .. ..... ...
.. .. ... .. .. .. ... . . . .. . ... .. .. .. . ... .. .... . ... .. ... ... . .. ... ... .. .. .. ..... ...
.. ... ... ... ... ... ... ... ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ..... .. . ... .... .. .. ..... ...
.. ... ... ... ... ... ... ... ... ... .. ... .. .. ... ... ... .. ... .. ... . ... ... ... . ... ... ... .. ... .. ... .. . . .. .. ... ... ... ... .. . ... . ... .. . .... ....
Fig. 3. 2C5 -configurations
3
Characterization of k-distance hereditary graphs
Definition 3.1 Let G be a graph and u and v be any two vertices of G. •
The dilation number of the pair {u, v} is given by δG (u, v) = DG (u, v) − dG (u, v).
•
The dilation number of the graph G denoted by δ(G) is the maximum dilation over all possible pairs of vertices in G, that is δ(G) = max{u,v}∈G δG (u, v).
•
The pairs of vertices with maximum dilation number consist in D(G) = {{u, v}/δG (u, v) = δ(G)}.
It is easy to verify that the class of k-distance hereditary graphs is closed under taking connected induced subgraphs. Consequently, every pair of non adjacent vertices u and v in a k-distance hereditary graph G will not have a dilation number more than k. For instance, when the distance dG (u, v) = 2, the cycle C4+k is forbidden for each k ≥ k. This configuration remains non
K. Meslem, M. Aïder / Electronic Notes in Discrete Mathematics 22 (2005) 253–259
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allowed if all the chords have a common endpoint. When the distance dG (u, v) is more than 2, we can build the forbidden configurations using the following operations : Definition 3.2 Let G1 = (V1 , E1 ) and G2 = (V2 , E2 ) be two graphs; and v1 ∈ V1 and v2 ∈ V2 two vertices such that |N (v1 )| = |N (v2 )| = 2. •
For each r ∈ N, the r-composition of G1 and G2 is the graph G1 ϕr G2 obtained by connecting G1 and G2 by a path P (v1 , v2 ) of length r.
•
G1 ϕ−1 G2 is the split composition of G1 and G2 with respect to the vertices v1 and v2 .
Definition 3.3 A k-configuration will designate any forbidden induced subgraph of a k-distance hereditary graph. This configuration is strict if its dilation number is equal to k. Otherwise, when the dilation number overtakes this value, the configuration is said large. It is not difficult to see that if we link a set of p 1C4+ki -configurations (1 ≤ ki < k), i = 1 . . . p and ( pi ki = k using the first operation above, we obtain a configuration which can not be contained in any k-distance hereditary graph. Such a graph is called a strict k-configuration of type (a) and can be given in the following form : G = 1C4+k1 ϕr1 1C4+k2 ϕr2 ...1C4+kp−1 ϕrp−1 1C4+kp where 2 ≤ p ≤ k; i = 1, ..., p; j = 1, ..., (p − 1); rj ∈ IN; i=p i=1 = k (Fig. 4).
.. .. ... .. .. .. . ... .... .. ... . .. ... ... . .. ... ... . .. ... ... . .. ... ... .. .. .. ...... ...
.. ... ... ... ... ... ... ... ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ..... .. . ... .... .. .. ...... ..
.. ... ... ... ... ... ... ... ... ... .. .. .. .. ... .. ... .. ... ... ... . ... ... ... . ... ... ... .. ... .. ... .. . . ... ... .. ... .. ... .... ... .. ... .. . .... ....
Fig. 4. 1C5 ϕr (1C5 ϕ−1 1C5 ) = 1C5 ϕr 2C5
In the other hand, a k-configuration of type (b) is any cycle-pair Cn {u, v} having no 1C4+k (k ≥ k) such that the length of the longest induced {u, v}path goes past the smallest one at least k unities (i.e DG (u, v) ≥ dG (u, v) + k).
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K. Meslem, M. Aïder / Electronic Notes in Discrete Mathematics 22 (2005) 253–259
Lemma 3.4 For each integer k, (k ≥ 2), a strict k-configuration of type (b) is any cycle C2p+k+2 {u, v} induced by a pair-cycle {u, v} such that: cd(C2p+k+2 ) = p and D(C2p+k+2 ) = {u, v}, where 2 ≤ p ≤ k Definition 3.5 A graph G is a k-configuration of type (a,b) if G is obtained by joining a ki -configuration of type (a) and a kj -configuration of type (b) with respect to the operation ϕr (r ∈ N ) such that Σi ki + Σj kj = k Bessedik [3] and Rautenbach [6] have characterized 3-distance hereditary graphs in terms of forbidden subgraphs. Among the 3-configurations that they provided, the 1C5 ϕr 1C5 ϕ−1 1C5 is a 3-configuration of type (a,b) where r ∈ N (See Fig. 4). Proposition 3.6 Let k be a positive integer. A strict k-configuration is one of the following graphs : (i) 1C4+k -configuration ; (ii) For each i = 1..p, j = 1..(p − 1), p = 2..k, the graph : 1C4+k1 ϕr1 1C4+k2 ϕr2 ...1C4+kp−1 ϕrp−1 1C4+kp ... 1 ≤ ki < k; Σi=p i=1 ki = k; rj ∈ N ∪ {−1}; (iii) All graphs obtained from the graph in (ii) by substituting each maximal cycle Cn {u, v} having a chord distance more than 1 and a dilation number k (k ≤ k) by a strict k -configuration of type (b).
References [1] M. A¨ıder, Almost Distance-Hereditary Graphs, Discrete Mathematics 242 (2002) 1–16. [2] H. J. Bandelt and H. M. Mulder, Distance Hereditary Graphs, J.Combin, Series (1986) B41, 182–208. [3] M. Bessedik, “Sur quelques Propri´et´es M´etriques des Graphes”, Magiter thesis, USTHB, Alger 1997. [4] S. Cicerone and G. Di Stefano, (k,+)-Distance-Hereditary Graphs, Journal of Discrete Algorithms, Volume 1, 3-4, 2003, 281–302. [5] E. Howorka, Characterization of Distance Hereditary Graphs, Quat. J. Math. Oxford(2) 28 (1977) 417–420.
K. Meslem, M. Aïder / Electronic Notes in Discrete Mathematics 22 (2005) 253–259
259
[6] D. Rautenbach, Graphs with Small Additive Stretch Number, Discussiones Mathematicae, Graph Theory 24 (2004) 291–301.
On an extension of distance-hereditary graphs Kahina Meslem 1 Facult´e des Math´ematiques U.S.T.H.B., B.P. 32 El Alia Algiers, Algeria
M´eziane A¨ıder 2 Facult´e des Math´ematiques U.S.T.H.B., B.P. 32 El Alia Algiers, Algeria
Abstract Given a simple and finite connected graph G, the distance dG (u, v) is the length of the shortest induced {u,v}-path linking the vertices u and v in G. Bandelt and Mulder [2] have characterized the class of distance hereditary graphs where the distance is preserved in each connected subgraph. In this paper, we are interested with the class of k-distance hereditary graphs (k ∈ N∗ ) which consists in a parametric extension of the distance-heredity notion. We allow the distance in each connected induced subgraph to increase by at most (k − 1) unities. We provide a characterization of k-distance hereditary graphs in terms of forbidden configurations for each k ∈ N∗ . Keywords: Distance heredity, Dilation number, Forbidden configurations.
1 2
Email: kahina [email protected] Email: [email protected]
1571-0653/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2005.06.056
254
1
K. Meslem, M. Aïder / Electronic Notes in Discrete Mathematics 22 (2005) 253–259
Introduction
The major problem that we meet in several networks consists of the inefficiency of transmitting messages between processors. Even if the sender and the destination are still connected, the messages are delivered with a certain delay. The distance is the main function which expresses the connectivity between the sites and the faulty tolerance in these networks. It is computed by means of shortest paths in networks. The k-distance hereditary graphs can be used to represent some networks communication. In fact, a network modeled as a k-distance hereditary graph (k ∈ N∗ ) is characterized as follows : if two sites have failed in transmission, as they remain connected, the distance between them in the faulty network is bounded by the distance in the original network plus an integer k −1, k ∈ N∗ . This means that a graph G belongs to k-distance hereditary class if and only if the length of each induced {u, v}-path is at most the distance in the original graph plus k − 1. Among this class, we can find the well known distance-hereditary graphs (k = 1) which are introduced by Howorka [5] and characterized in several ways by Bandelt and Mulder [2] using both algorithmic and metric aspect. Moreover, A¨ıder [1] has introduced and studied the 2-distance hereditary graphs (he called almost-distance hereditary graphs). Using the characterization of these graphs, other both combinatorial and metric proprieties were provided. Cicerone and Di Stefano [4] have used the notion of twin graph to characterize the k-distance hereditary graphs and have stated the coNP-completeness of the recognizing problem in this class. In this paper, we characterize k-distance hereditary graphs in terms of forbidden subgraphs using the chord distance in cycles and the dilation number of a graph. We provide all forbidden configurations for each integer k ≥ 1.
2
Preliminaries definitions
We only consider finite, simple loopless and undirected graphs G=(V ,E) where V is the vertex set and E is the edge set. A subgraph of G is a graph having all its vertices and edges in G. The neighborhood of a vertex u denoted by N (u) consists in all the vertices v which are adjacent to u. Given a subset S of V , the induced subgraph < S > of G is the maximal subgraph of G with vertex set S. An induced path of G is a non redundant connection of a pair of vertices. The length of a shortest path between two vertices u and v is called distance and is denoted by dG (u, v) ; moreover the length of a longest induced path between the same vertices is denoted by DG (u, v).
K. Meslem, M. Aïder / Electronic Notes in Discrete Mathematics 22 (2005) 253–259
255
A cycle Cn is a path (x1 , x2 , ..., xn , x1 ). For convenience, the (x, y) part of a path P is denoted by P (x.y). A chord of a cycle is an edge that joins non consecutive vertices of Cn . The chord distance of a cycle denoted by cd(Cn ) is defined to be the minimum number of consecutive vertices in Cn such that every chord of Cn is incident to some of such vertices. However, each cycle on n vertices with chord distance equal either to 0 or to 1 will be referred to as a 1Cn -configuration (see Fig. 1, where dashed lines represent eventual chords in the cycle). .. .. ... .. .. .. ... . . .. .. ... .. .. .. ... . . . .. . ... .. .. ... . ... .. ... ... .. .. .. ...... ...
Fig. 1. 1C5 -configurations
A pair {u, v} of non adjacent vertices is called cycle-pair if there exists a shortest induced {u, v}-path P and a longest induced {u, v}-path Q such that P ∩ Q = {u, v}. So, P ∪ Q induce a cycle denoted C{u, v}. The split composition of the graphs G1 = (V1 ∪ m1 , E1 ) and G2 = (V2 ∪ m2 , E2 ) with respect to m1 and m2 is the graph G1 ∗ G2 having vertex set V = V1 ∪ V2 and edge set E = E1 ∪ E2 ∪ {(u, v)/u ∈ N (m1 ); v ∈ N (m2 )}. (See Fig. 2).
m1
m2
Fig. 2. The split composition of two C5 with respect to m1 and m2
Definition 2.1 Let k be a positive integer. A graph G is k-distance hereditary if and only if for any connected induced subgraph H of G we have: ∀u, v ∈ H, dH (u, v) ≤ dG (u, v) + (k − 1) A¨ıder [1] has given a characterization of 2-distance hereditary graphs by providing all forbidden configurations, which are the 1Cn configuration for each
256
K. Meslem, M. Aïder / Electronic Notes in Discrete Mathematics 22 (2005) 253–259
n ≥ 6 and the 2C5 -configurations. In Fig. 3 (and in all our figures) a dashed line represents an eventual edge and a dotted line represents a path, eventually empty (in this case, the end vertices coincide). Other characterizations in terms of forbidden subgraphs involving chord distance proprieties were established for each k-distance hereditary graphs (k ≥ 1). For instance, Cicerone and Di Stefano [4] showed that a graph G is kdistance hereditary if and only if the chord distance of each cycle Cn , with (n > k + 5), is more than n−k+1
− 1 in the twin graph G∗ . The twin graph 2 ∗ G is obtained by duplicating every vertex (by a false twin) of G. The implicit form of this result incites us to check for the forbidden subgraphs of k-distance hereditary graphs. That is the main target of the following section.
.. .. ... .. .. .. ... . . . .. . ... .. .. .. ... . .. .... . ... .. ... ... . .. ... ... .. .. .. ..... ...
.. .. ... .. .. .. ... . . . .. . ... .. .. .. . ... .. .... . ... .. ... ... . .. ... ... .. .. .. ..... ...
.. ... ... ... ... ... ... ... ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ..... .. . ... .... .. .. ..... ...
.. ... ... ... ... ... ... ... ... ... .. ... .. .. ... ... ... .. ... .. ... . ... ... ... . ... ... ... .. ... .. ... .. . . .. .. ... ... ... ... .. . ... . ... .. . .... ....
Fig. 3. 2C5 -configurations
3
Characterization of k-distance hereditary graphs
Definition 3.1 Let G be a graph and u and v be any two vertices of G. •
The dilation number of the pair {u, v} is given by δG (u, v) = DG (u, v) − dG (u, v).
•
The dilation number of the graph G denoted by δ(G) is the maximum dilation over all possible pairs of vertices in G, that is δ(G) = max{u,v}∈G δG (u, v).
•
The pairs of vertices with maximum dilation number consist in D(G) = {{u, v}/δG (u, v) = δ(G)}.
It is easy to verify that the class of k-distance hereditary graphs is closed under taking connected induced subgraphs. Consequently, every pair of non adjacent vertices u and v in a k-distance hereditary graph G will not have a dilation number more than k. For instance, when the distance dG (u, v) = 2, the cycle C4+k is forbidden for each k ≥ k. This configuration remains non
K. Meslem, M. Aïder / Electronic Notes in Discrete Mathematics 22 (2005) 253–259
257
allowed if all the chords have a common endpoint. When the distance dG (u, v) is more than 2, we can build the forbidden configurations using the following operations : Definition 3.2 Let G1 = (V1 , E1 ) and G2 = (V2 , E2 ) be two graphs; and v1 ∈ V1 and v2 ∈ V2 two vertices such that |N (v1 )| = |N (v2 )| = 2. •
For each r ∈ N, the r-composition of G1 and G2 is the graph G1 ϕr G2 obtained by connecting G1 and G2 by a path P (v1 , v2 ) of length r.
•
G1 ϕ−1 G2 is the split composition of G1 and G2 with respect to the vertices v1 and v2 .
Definition 3.3 A k-configuration will designate any forbidden induced subgraph of a k-distance hereditary graph. This configuration is strict if its dilation number is equal to k. Otherwise, when the dilation number overtakes this value, the configuration is said large. It is not difficult to see that if we link a set of p 1C4+ki -configurations (1 ≤ ki < k), i = 1 . . . p and ( pi ki = k using the first operation above, we obtain a configuration which can not be contained in any k-distance hereditary graph. Such a graph is called a strict k-configuration of type (a) and can be given in the following form : G = 1C4+k1 ϕr1 1C4+k2 ϕr2 ...1C4+kp−1 ϕrp−1 1C4+kp where 2 ≤ p ≤ k; i = 1, ..., p; j = 1, ..., (p − 1); rj ∈ IN; i=p i=1 = k (Fig. 4).
.. .. ... .. .. .. . ... .... .. ... . .. ... ... . .. ... ... . .. ... ... . .. ... ... .. .. .. ...... ...
.. ... ... ... ... ... ... ... ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ..... .. . ... .... .. .. ...... ..
.. ... ... ... ... ... ... ... ... ... .. .. .. .. ... .. ... .. ... ... ... . ... ... ... . ... ... ... .. ... .. ... .. . . ... ... .. ... .. ... .... ... .. ... .. . .... ....
Fig. 4. 1C5 ϕr (1C5 ϕ−1 1C5 ) = 1C5 ϕr 2C5
In the other hand, a k-configuration of type (b) is any cycle-pair Cn {u, v} having no 1C4+k (k ≥ k) such that the length of the longest induced {u, v}path goes past the smallest one at least k unities (i.e DG (u, v) ≥ dG (u, v) + k).
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K. Meslem, M. Aïder / Electronic Notes in Discrete Mathematics 22 (2005) 253–259
Lemma 3.4 For each integer k, (k ≥ 2), a strict k-configuration of type (b) is any cycle C2p+k+2 {u, v} induced by a pair-cycle {u, v} such that: cd(C2p+k+2 ) = p and D(C2p+k+2 ) = {u, v}, where 2 ≤ p ≤ k Definition 3.5 A graph G is a k-configuration of type (a,b) if G is obtained by joining a ki -configuration of type (a) and a kj -configuration of type (b) with respect to the operation ϕr (r ∈ N ) such that Σi ki + Σj kj = k Bessedik [3] and Rautenbach [6] have characterized 3-distance hereditary graphs in terms of forbidden subgraphs. Among the 3-configurations that they provided, the 1C5 ϕr 1C5 ϕ−1 1C5 is a 3-configuration of type (a,b) where r ∈ N (See Fig. 4). Proposition 3.6 Let k be a positive integer. A strict k-configuration is one of the following graphs : (i) 1C4+k -configuration ; (ii) For each i = 1..p, j = 1..(p − 1), p = 2..k, the graph : 1C4+k1 ϕr1 1C4+k2 ϕr2 ...1C4+kp−1 ϕrp−1 1C4+kp ... 1 ≤ ki < k; Σi=p i=1 ki = k; rj ∈ N ∪ {−1}; (iii) All graphs obtained from the graph in (ii) by substituting each maximal cycle Cn {u, v} having a chord distance more than 1 and a dilation number k (k ≤ k) by a strict k -configuration of type (b).
References [1] M. A¨ıder, Almost Distance-Hereditary Graphs, Discrete Mathematics 242 (2002) 1–16. [2] H. J. Bandelt and H. M. Mulder, Distance Hereditary Graphs, J.Combin, Series (1986) B41, 182–208. [3] M. Bessedik, “Sur quelques Propri´et´es M´etriques des Graphes”, Magiter thesis, USTHB, Alger 1997. [4] S. Cicerone and G. Di Stefano, (k,+)-Distance-Hereditary Graphs, Journal of Discrete Algorithms, Volume 1, 3-4, 2003, 281–302. [5] E. Howorka, Characterization of Distance Hereditary Graphs, Quat. J. Math. Oxford(2) 28 (1977) 417–420.
K. Meslem, M. Aïder / Electronic Notes in Discrete Mathematics 22 (2005) 253–259
259
[6] D. Rautenbach, Graphs with Small Additive Stretch Number, Discussiones Mathematicae, Graph Theory 24 (2004) 291–301.