^{1}

^{2}

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An interval

We discuss finite simple graphs which are variations on the well-studied class of interval graphs. Interval graphs have been extensively studied and characterized, and fast algorithms for various problems such as clique number, chromatic number, dominating sets, and many others have been developed. Indeed it is at this point difficult to give a thorough list of references or a single reference with a sufficient representation of even recent work done on or with interval graphs and several of their variants. The variant we consider is as follows. Suppose

If

Let

The complete list of forbidden induced subgraphs for

We think our results are an illustration of how the complexity of interval

A

Let

A graph

A graph

An

If

A graph is

The class of interval

Note that this is why, in trying to represent graph

The following facts and the next lemma will provide the basis for our characterization:

a

proper interval

interval

any subgraph obtained from vertex deletion of an interval

Each graph in the infinite family in Figure

An infinite family of forbidden induced subgraphs for proper interval

Assume for contradiction that there is a proper representation for

The vertical line signifies the contradiction in the proof of Lemma

Our arguments for the proof of the main theorem will be facilitated by the notion of trapping which we now define and then prove a lemma about the structure of a proper interval

Now

Representations used in Lemma

The forbidden induced subgraphs for

Since interval

Let

Let

Now without loss of generality assume that

We are now ready to prove the main result.

Let

Assume that

Assume for contradiction that

This gives us

Because

Now let us assume that the path from

Now assume that

Label the path from

Now assume that there is only one

Since

Consider the first case. If

Now consider

Because

The interval for

Let us first consider the subgraph of

Now consider the structure of the graph around the vertices

Next we show that

Now we can see that if both ends of

In each case we found that a graph from Figure

If we combine Theorem

A

Combining the above corollary with Theorem

A chordal uniquely

A

The unit interval graphs are equivalent to the proper interval graphs, and they are further equivalent to the

A

A graph

Let

Let

Combining Lemma

Let

In this paper, we have given a characterization of proper interval

A unit (proper) interval

However we conjecture that the absence of any graph from the infinite family of graphs in Figure

If we consider proper interval

Some forbidden induced subgraphs with

The authors declare that there is no conflict of interests regarding the publication of this paper.