Related papers: Clique Cover Width and Clique Sum
The clique vector $\mathfrak{c}(G)$ of a graph $G$ is the sequence $(c_1, c_2, \ldots,c_d)$ in $\mathbb{N}^d$, where $c_i$ is the number of cliques in $G$ with $i$ vertices and $d$ is the largest cardinality of a clique in $G$. In this…
An edge clique cover of a graph is a set of cliques that covers all edges of the graph. We generalize this concept to "$K_t$ clique cover", i.e. a set of cliques that covers all complete subgraphs on $t$ vertices of the graph, for every $t…
Clique-width is a graph invariant that has been widely studied in combinatorics and computer science. However, computing the clique-width of a graph is an intricate problem, the exact clique-width is not known even for very small graphs. We…
Clique-width is a well-studied graph parameter owing to its use in understanding algorithmic tractability: if the clique-width of a graph class ${\cal G}$ is bounded by a constant, a wide range of problems that are NP-complete in general…
The bandwidth of a graph is the labeling of vertices with minimum maximum edge difference. For many graph families this is NP-complete. A classic result computes the bandwidth for the hypercube. We generalize this result to give sharp lower…
Let $\cal C$ be a clique covering for $E(G)$ and let $v$ be a vertex of $G$. The valency of vertex $v$ (with respect to $\cal C$), denoted by $val_{\cal C}(v)$, is the number of cliques in $\cal C$ containing $v$. The local clique cover…
The bandwidth of a graph G is the minimum of the maximum difference between adjacent labels when the vertices have distinct integer labels. We provide a polynomial algorithm to produce an optimal bandwidth labeling for graphs in a special…
The ordinary generating function of the number of complete subgraphs (cliques) of $G$, denoted by $C(G,x)$, is called the The clique polynomial of the graph $G$. In this paper, we first introduce some \emph{clique} incidence matrices…
In this paper we prove that the recursive (Knill) dimension of the join of two graphs has a simple formula in terms of the dimensions of the component graphs: $\mathrm{dim\,} (G_1+G_2) = 1 +\mathrm{dim\,} G_1+ \mathrm{dim\,} G_2$. We use…
In this paper we prove that the inductively defined graph dimension has a simple additive property under the join operation. The dimension of the join of two simple graphs is one plus the sum of the dimensions of the component graphs:…
Clique-width is a well-studied graph parameter. For graphs of bounded clique-width, many problems that are NP-hard in general can be polynomial-time solvable. The fact motivates several studies to investigate whether the clique-width of…
We give an $O(n^4)$ algorithm to find a minimum clique cover of a (bull, $C_4$)-free graph, or equivalently, a minimum colouring of a (bull, $2K_2$)-free graph, where $n$ is the number of vertices of the graphs.
It takes $n^2/4$ cliques to cover all the edges of a complete bipartite graph $K_{n/2,n/2}$, but how many cliques does it take to cover all the edges of a graph $G$ if $G$ has no $K_{t,t}$ induced subgraph? We prove that $O(|G|^{2-1/(2t)})$…
For each m>=1 and k>=2, we construct a graph G=(V,E) with \omega(G)=m such that max_{1\leq i\leq k} \omega(G[V_i])=m for arbitrary partition V=V_1\cup...\cup V_k, where \omega(G) is the clique number of G and G[V_i] is the induced subgraph…
Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G, x)=\sum_{i=1}^n d(G,i) x^i$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. For two graphs $G$ and $H$, let $\mathcal{C} =…
Let $ G=(V,E) $ be a simple graph of order $ n $ and size $ m $. A connected edge cover set of a graph is a subset $S$ of edges such that every vertex of the graph is incident to at least one edge of $S$ and the subgraph induced by $S$ is…
Let $B$ and $R$ be two simple graphs with vertex set $V$, and let $G(B,R)$ be the simple graph with vertex set $V$, in which two vertices are adjacent if they are adjacent in at least one of $B$ and $R$. For $X \subseteq V$, we denote by…
We prove that edge contractions do not preserve the property that a set of graphs has bounded clique-width. This property is preserved by contractions of edges, one end of which is a vertex of degree 2.
We define the cover number of a graph $G$ by a graph class $\mathcal P$ as the minimum number of graphs of class $\mathcal P$ required to cover the edge set of $G$. Taking inspiration from a paper by Harary, Hsu and Miller, we find an exact…
A clique transversal in a graph is a set of vertices intersecting all maximal cliques. The problem of determining the minimum size of a clique transversal has received considerable attention in the literature. In this paper, we initiate the…