Minimal split completions

We study the problem of adding an inclusion minimal set of edges to a given arbitrary graph so that the resulting graph is a split graph, called a minimal split completion of the input graph. Minimal completions of arbitrary graphs into chordal and interval graphs have been studied previously, and new results have been added recently. We extend these previous results to split graphs by giving a linear-time algorithm for computing minimal split completions. We also give two characterizations of minimal split completions, which lead to a linear time algorithm for extracting a minimal split completion from any given split completion.We prove new properties of split graph that are both useful for our algorithms and interesting on their own. First, we present a new way of partitioning the vertices of a split graph uniquely into three subsets. Second, we prove that split graphs have the following property: given two split graphs on the same vertex set where one is a subgraph of the other, there is a sequence of edges that can be removed from the larger to obtain the smaller such that after each edge removal the modified graph is split.     

Minimal comparability completions of arbitrary graphs

A transitive orientation of an undirected graph is an assignment of directions to its edges so that these directed edges represent a transitive relation between the vertices of the graph. Not every graph has a transitive orientation, but every graph can be turned into a graph that has a transitive orientation, by adding edges. We study the problem of adding an inclusion minimal set of edges to an arbitrary graph so that the resulting graph is transitively orientable. We show that this problem can be solved in polynomial time, and we give a surprisingly simple algorithm for it. We use a vertex incremental approach in this algorithm, and we also give a more general result that describes graph classes Π for which Π completion of arbitrary graphs can be achieved through such a vertex incremental approach.     

Total domination changing and stable graphs upon vertex removal

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. A graph is total domination vertex removal stable if the removal of an arbitrary vertex leaves the total domination number unchanged. On the other hand, a graph is total domination vertex removal changing if the removal of an arbitrary vertex changes the total domination number. In this paper, we study total domination vertex removal changing and stable graphs.     

Finite common coverings of pairs of regular graphs

The graph G is a covering of the graph H if there exists a (projection) map p from the vertex set of G to the vertex set of H which induces a one-to-one correspondence between the vertices adjacent to v in G and the vertices adjacent to p(v) in H, for every vertex v of G. We show that for any two finite regular graphs G and H of the same degree, there exists a finite graph K that is simultaneously a covering both of G and H. The proof uses only Hall's theorem on 1-factors in regular bipartite graphs.     

Fully dynamic recognition algorithm and certificate for directed cographs

This paper presents an optimal fully dynamic recognition algorithm for directed cographs. Given the modular decomposition tree of a directed cograph G, the algorithm supports arc and vertex modification (insertion or deletion) in O(d) time where d is the number of arcs involved in the operation. Moreover, if the modified graph remains a directed cograph, the modular decomposition tree is updated; otherwise, a certificate is returned within the same complexity.     

A polynomial-time algorithm for the paired-domination problem on permutation graphs

A set S of vertices in a graph H=(V,E) with no isolated vertices is a paired-dominating set of H if every vertex of H is adjacent to at least one vertex in S and if the subgraph induced by S contains a perfect matching. Let G be a permutation graph and π be its corresponding permutation. In this paper we present an O(mn) time algorithm for finding a minimum cardinality paired-dominating set for a permutation graph G with n vertices and m edges.     

Generating and Enumerating Digitally Convex Sets of Trees

A set S of vertices in a graph G with vertex set V is digitally convex if for every vertex \(v \in V\), \(N[v] \subseteq N[S]\) implies \(v \in S\). We show that a vertex belongs to at most half of the digitally convex sets of a graph. Moreover, a vertex belongs to exactly half of the digitally convex sets if and only if it is simplicial. An algorithm that generates all digitally convex sets of a tree is described and sharp upper and lower bounds for the number of digitally convex sets of a tree are obtained. A closed formula for the number of digitally convex sets of a path is derived. It is shown how a binary cotree of a cograph can be used to enumerate its digitally convex sets in linear time.     

A finiteness theorem for primal extensions

A set W⊆V(G) is called homogeneous in a graph G if 2?|W|?|V(G)|-1, and N(x)?W=N(y)?W for each x,y∈W. A graph without homogeneous sets is called prime. A graph H is called a (primal) extension of a graph G if G is an induced subgraph of H, and H is a prime graph. An extension H of G is minimal if there are no extensions of G in the set ISub(H)?{H}. We denote by Ext(G) the set of all minimal extensions of a graph G.We investigate the following problem: find conditions under which Ext(G) is a finite set. The main result of Giakoumakis (Discrete Math. 177 (1997) 83-97) is the following sufficient condition.

Theorem. If every homogeneous set of G has exactly two vertices thenExt(G)is a finite set.     

Total domination critical and stable graphs upon edge removal

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. A graph is total domination edge critical if the removal of any arbitrary edge increases the total domination number. On the other hand, a graph is total domination edge stable if the removal of any arbitrary edge has no effect on the total domination number. In this paper, we characterize total domination edge critical graphs. We also investigate various properties of total domination edge stable graphs.     

Geodetic and Steiner geodetic sets in 3-Steiner distance hereditary graphs

Let G be a connected graph and S⊆V(G). Then the Steiner distance of S, denoted by d_G(S), is the smallest number of edges in a connected subgraph of G containing S. Such a subgraph is necessarily a tree called a Steiner tree for S. The Steiner interval for a set S of vertices in a graph, denoted by I(S) is the union of all vertices that belong to some Steiner tree for S. If S={u,v}, then I(S) is the interval I[u,v] between u and v. A connected graph G is 3-Steiner distance hereditary (3-SDH) if, for every connected induced subgraph H of order at least 3 and every set S of three vertices of H, d_H(S)=d_G(S). The eccentricity of a vertex v in a connected graph G is defined as e(v)=max{d(v,x)|x∈V(G)}. A vertex v in a graph G is a contour vertex if for every vertex u adjacent with v, e(u)?e(v). The closure of a set S of vertices, denoted by I[S], is defined to be the union of intervals between pairs of vertices of S taken over all pairs of vertices in S. A set of vertices of a graph G is a geodetic set if its closure is the vertex set of G. The smallest cardinality of a geodetic set of G is called the geodetic number of G and is denoted by g(G). A set S of vertices of a connected graph G is a Steiner geodetic set for G if I(S)=V(G). The smallest cardinality of a Steiner geodetic set of G is called the Steiner geodetic number of G and is denoted by sg(G). We show that the contour vertices of 3-SDH and HHD-free graphs are geodetic sets. For 3-SDH graphs we also show that g(G)?sg(G). An efficient algorithm for finding Steiner intervals in 3-SDH graphs is developed.     

A vertex incremental approach for maintaining chordality

For a chordal graph G=(V,E), we study the problem of whether a new vertex u∉V and a given set of edges between u and vertices in V can be added to G so that the resulting graph remains chordal. We show how to resolve this efficiently, and at the same time, if the answer is no, specify a maximal subset of the proposed edges that can be added along with u, or conversely, a minimal set of extra edges that can be added in addition to the given set, so that the resulting graph is chordal. In order to do this, we give a new characterization of chordal graphs and, for each potential new edge uv, a characterization of the set of edges incident to u that also must be added to G along with uv. We propose a data structure that can compute and add each such set in O(n) time. Based on these results, we present an algorithm that computes both a minimal triangulation and a maximal chordal subgraph of an arbitrary input graph in O(nm) time, using a totally new vertex incremental approach. In contrast to previous algorithms, our process is on-line in that each new vertex is added without reconsidering any choice made at previous steps, and without requiring any knowledge of the vertices that might be added subsequently.     

The geodetic number of the lexicographic product of graphs

A set S of vertices of a graph G is a geodetic set if every vertex of G lies in an interval between two vertices from S. The size of a minimum geodetic set in G is the geodetic number g(G) of G. We find that the geodetic number of the lexicographic product G°H for a non-complete graph H lies between 2 and 3g(G). We characterize the graphs G and H for which g(G°H)=2, as well as the lexicographic products T°H that enjoy g(T°H)=3g(G), when T is isomorphic to a tree. Using a new concept of the so-called geodominating triple of a graph G, a formula that expresses the exact geodetic number of G°H is established, where G is an arbitrary graph and H a non-complete graph.     

Vertex Coverings by Coloured Induced Graphs—Frames and Umbrellas

A graph G homogeneously embeds in a graph H if for every vertex x of G and every vertex y of H there is an induced copy of G in H with x at y. The graph G uniformly embeds in H if for every vertex y of H there is an induced copy of G in H containing y. For positive integer k, let f_k(G) (respectively, g_k(G)) be the minimum order of a graph H whose edges can be k-coloured such that for each colour, G homogeneously embeds (respectively, uniformly embeds) in the graph given by V(H) and the edges of that colour. We investigate the values f₂(G) and g₂(G) for special classes of G, in particular when G is a star or balanced complete bipartite graph. Then we investigate f_k(G) and g_k(G) when k ≥ 3 and G is a complete graph.     

Minimal separating sets of maximum size

A connected graph G can be disconnected or reduced to a single vertex by removing an appropriate subset of the vertex set V(G), and can be disconnected by removing a suitable subset of the edge set E(G). Attention has usually been centered on separating sets having minimum cardinality, and parameters called the vertex connectivity and the edge connectivity defined. These classical concepts are generalized by using separating sets which are minimal. By considering the maximum as well as the minimum cardinality of such sets, one defines vertex and edge connectivity parameters. Sharp upper bounds are established for these numbers and their values computed for certain classes of graphs. An analogue of Whitney's theorem on connectivity is obtained. Parameters are also defined for minimal separating sets consisting of a mixture of vertices and edges, and these are shown to depend on the maximum and minimum values of the vertex and edge connectivity parameters.     

Graham's pebbling conjecture on products of cycles

Chung defined a pebbling move on a graph G to be the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. The pebbling number of a connected graph is the smallest number f(G) such that any distribution of f(G) pebbles on G allows one pebble to be moved to any specified, but arbitrary vertex by a sequence of pebbling moves. Graham conjectured that for any connected graphs G and H, f(G×H)≤ f(G)f(H). We prove Graham's conjecture when G is a cycle for a variety of graphs H, including all cycles. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 141–154, 2003     

On the bondage number of middle graphs

Let G = (V (G),E(G)) be a simple graph. A subset S of V (G) is a dominating set of G if, for any vertex v ∈ V (G) — S, there exists some vertex u ∈ S such that uv ∈ E(G). The domination number, denoted by γ(G), is the cardinality of a minimal dominating set of G. There are several types of domination parameters depending upon the nature of domination and the nature of dominating set. These parameters are bondage, reinforcement, strong-weak domination, strong-weak bondage numbers. In this paper, we first investigate the strong-weak domination number of middle graphs of a graph. Then several results for the bondage, strong-weak bondage number of middle graphs are obtained.     

Circular vertex arboricity

The vertex arboricity va(G) of graph G is defined as the minimum of subsets in a partition of the vertex set of G so that each subset induces an acyclic subgraph and has been widely studied. We define the concept of circular vertex arboricity va_c(G) of graph G, which is a natural generalization of vertex arboricity. We give some basic properties of circular vertex arboricity and study the circular vertex arboricity of planar graphs.     

Partitioning 2-edge-colored graphs by monochromatic paths and cycles

We present results on partitioning the vertices of 2-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of Sárközy: the vertex set of every 2-edge-colored graph can be partitioned into at most 2α(G) monochromatic cycles, where α(G) denotes the independence number of G. Another direction, emerged recently from a conjecture of Schelp, is to consider colorings of graphs with given minimum degree. We prove that apart from o(|V (G)|) vertices, the vertex set of any 2-edge-colored graph G with minimum degree at least \(\tfrac{{(1 + \varepsilon )3|V(G)|}} {4}\) can be covered by the vertices of two vertex disjoint monochromatic cycles of distinct colors. Finally, under the assumption that \(\bar G\) does not contain a fixed bipartite graph H, we show that in every 2-edge-coloring of G, |V (G)| ? c(H) vertices can be covered by two vertex disjoint paths of different colors, where c(H) is a constant depending only on H. In particular, we prove that c(C ₄)=1, which is best possible.     

On the remoteness function in median graphs

A profile on a graph G is any nonempty multiset whose elements are vertices from G. The corresponding remoteness function associates to each vertex x∈V(G) the sum of distances from x to the vertices in the profile. Starting from some nice and useful properties of the remoteness function in hypercubes, the remoteness function is studied in arbitrary median graphs with respect to their isometric embeddings in hypercubes. In particular, a relation between the vertices in a median graph G whose remoteness function is maximum (antimedian set of G) with the antimedian set of the host hypercube is found. While for odd profiles the antimedian set is an independent set that lies in the strict boundary of a median graph, there exist median graphs in which special even profiles yield a constant remoteness function. We characterize such median graphs in two ways: as the graphs whose periphery transversal number is 2, and as the graphs with the geodetic number equal to 2. Finally, we present an algorithm that, given a graph G on n vertices and m edges, decides in O(mlogn) time whether G is a median graph with geodetic number 2.     

Distance-biregular graphs with 2-valent vertices and distance-regular line graphs

A graph G is called distance-regularized if each vertex of G admits an intersection array. It is known that every distance-regularized graph is either distance-regular (DR) or distance-biregular (DBR). Note that DBR means that the graph is bipartite and the vertices in the same color class have the same intersection array. A (k, g)-graph is a k-regular graph with girth g and with the minimum possible number of vertices consistent with these properties. Biggs proved that, if the line graph L(G) is distance-transitive, then G is either K_1,n or a (k, g)-graph. This result is generalized to DR graphs by showing that the following are equivalent: (1) L(G) is DR and G ≠ K_1,n for n ≥ 2, (2) G and L(G) are both DR, (3) subdivision graph S(G) is DBR, and (4) G is a (k, g)-graph. This result is used to show that a graph S is a DBR graph with 2-valent vertices iff S = K_2,′ or S is the subdivision graph of a (k, g)-graph. Let G⁽²⁾ be the graph with vertex set that of G and two vertices adjacent if at distance two in G. It is shown that for a DBR graph G, G⁽²⁾ is two DR graphs. It is proved that a DR graph H without triangles can be obtained as a component of G⁽²⁾ if and only if it is a (k, g)-graph with g ≥ 4.